Electromagnetic mass

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https://en.wikipedia.org/wiki/Electromagnetic_mass 

Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for some time also considered as a dynamical explanation of inertial mass per se. Today, the relation of mass, momentum, velocity, and all forms of energy – including electromagnetic energy – is analyzed on the basis of Albert Einstein's special relativity and mass–energy equivalence. As to the cause of mass of elementary particles, the Higgs mechanism in the framework of the relativistic Standard Model is currently used. However, some problems concerning the electromagnetic mass and self-energy of charged particles are still studied.

Charged particles

Rest mass and energy

It was recognized by J. J. Thomson in 1881 that a charged sphere moving in a space filled with a medium of a specific inductive capacity (the electromagnetic aether of James Clerk Maxwell), is harder to set in motion than an uncharged body. (Similar considerations were already made by George Gabriel Stokes (1843) with respect to hydrodynamics, who showed that the inertia of a body moving in an incompressible perfect fluid is increased.) So due to this self-induction effect, electrostatic energy behaves as having some sort of momentum and "apparent" electromagnetic mass, which can increase the ordinary mechanical mass of the bodies, or in more modern terms, the increase should arise from their electromagnetic self-energy. This idea was worked out in more detail by Oliver Heaviside (1889), Thomson (1893), George Frederick Charles Searle (1897), Max Abraham (1902), Hendrik Lorentz (1892, 1904), and was directly applied to the electron by using the Abraham–Lorentz force. Now, the electrostatic energy ${\displaystyle E_{\mathrm {em} }}$ and mass ${\displaystyle m_{\mathrm {em} }}$ of an electron at rest was calculated to be

${\displaystyle E_{\mathrm {em} }={\frac {1}{2}}{\frac {e^{2}}{a}},\qquad m_{\mathrm {em} }={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}}$

where ${\displaystyle e}$ is the charge, uniformly distributed on the surface of a sphere, and ${\displaystyle a}$ is the classical electron radius, which must be nonzero to avoid infinite energy accumulation. Thus the formula for this electromagnetic energy–mass relation is

${\displaystyle m_{\mathrm {em} }={\frac {4}{3}}{\frac {E_{\mathrm {em} }}{c^{2}}}}$

This was discussed in connection with the proposal of the electrical origin of matter, so Wilhelm Wien (1900), and Max Abraham (1902), came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass. Wien stated, that if it is assumed that gravitation is an electromagnetic effect too, then there has to be a proportionality between electromagnetic energy, inertial mass, and gravitational mass. When one body attracts another one, the electromagnetic energy store of gravitation is according to Wien diminished by the amount (where ${\displaystyle M}$ is the attracted mass, ${\displaystyle G}$ the gravitational constant, ${\displaystyle r}$ the distance):

${\displaystyle G={\frac {{\frac {4}{3}}{\frac {E_{\mathrm {em} }}{c^{2}}}M}{r}}}$

Henri Poincaré in 1906 argued that when mass is in fact the product of the electromagnetic field in the aether – implying that no "real" mass exists – and because matter is inseparably connected with mass, then also matter doesn't exist at all and electrons are only concavities in the aether.

Mass and speed

Thomson and Searle

Thomson (1893) noticed that electromagnetic momentum and energy of charged bodies, and therefore their masses, depend on the speed of the bodies as well. He wrote:

[p. 21] When in the limit v = c, the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain constant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light.

In 1897, Searle gave a more precise formula for the electromagnetic energy of charged sphere in motion:

${\displaystyle E_{\mathrm {em} }^{v}=E_{\mathrm {em} }\left[{\frac {1}{\beta }}\ln {\frac {1+\beta }{1-\beta }}-1\right],\qquad \beta ={\frac {v}{c}},}$

and like Thomson he concluded:

... when v = c the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.

Longitudinal and transverse mass

Predictions of speed dependence of transverse electromagnetic mass according to the theories of Abraham, Lorentz, and Bucherer.

From Searle's formula, Walter Kaufmann (1901) and Abraham (1902) derived the formula for the electromagnetic mass of moving bodies:

${\displaystyle m_{L}={\frac {3}{4}}\cdot m_{\mathrm {em} }\cdot {\frac {1}{\beta ^{2}}}\left[-{\frac {1}{\beta }}\ln \left({\frac {1+\beta }{1-\beta }}\right)+{\frac {2}{1-\beta ^{2}}}\right]}$

However, it was shown by Abraham (1902), that this value is only valid in the longitudinal direction ("longitudinal mass"), i.e., that the electromagnetic mass also depends on the direction of the moving bodies with respect to the aether. Thus Abraham also derived the "transverse mass":

${\displaystyle m_{T}={\frac {3}{4}}\cdot m_{\mathrm {em} }\cdot {\frac {1}{\beta ^{2}}}\left[\left({\frac {1+\beta ^{2}}{2\beta }}\right)\ln \left({\frac {1+\beta }{1-\beta }}\right)-1\right]}$

On the other hand, already in 1899 Lorentz assumed that the electrons undergo length contraction in the line of motion, which leads to results for the acceleration of moving electrons that differ from those given by Abraham. Lorentz obtained factors of ${\displaystyle k^{3}\varepsilon }$ parallel to the direction of motion and ${\displaystyle k\varepsilon }$ perpendicular to the direction of motion, where ${\displaystyle k={\sqrt {1-v^{2}/c^{2}}}}$ and ${\displaystyle \varepsilon }$ is an undetermined factor. Lorentz expanded his 1899 ideas in his famous 1904 paper, where he set the factor ${\displaystyle \varepsilon }$ to unity, thus:

${\displaystyle m_{L}={\frac {m_{\mathrm {em} }}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{3}}},\quad m_{T}={\frac {m_{\mathrm {em} }}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$,

So, eventually Lorentz arrived at the same conclusion as Thomson in 1893: no body can reach the speed of light because the mass becomes infinitely large at this velocity.

Additionally, a third electron model was developed by Alfred Bucherer and Paul Langevin, in which the electron contracts in the line of motion, and expands perpendicular to it, so that the volume remains constant. This gives:

${\displaystyle m_{L}={\frac {m_{\mathrm {em} }\left(1-{\frac {1}{3}}{\frac {v^{2}}{c^{2}}}\right)}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{8/3}}},\quad m_{T}={\frac {m_{\mathrm {em} }}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{2/3}}}}$

Kaufmann's experiments

The predictions of the theories of Abraham and Lorentz were supported by the experiments of Walter Kaufmann (1901), but the experiments were not precise enough to distinguish between them. In 1905 Kaufmann conducted another series of experiments (Kaufmann–Bucherer–Neumann experiments) which confirmed Abraham's and Bucherer's predictions, but contradicted Lorentz's theory and the "fundamental assumption of Lorentz and Einstein", i.e., the relativity principle. In the following years experiments by Alfred Bucherer (1908), Gunther Neumann (1914) and others seemed to confirm Lorentz's mass formula. It was later pointed out that the Bucherer–Neumann experiments were also not precise enough to distinguish between the theories – it lasted until 1940 when the precision required was achieved to eventually prove Lorentz's formula and to refute Abraham's by these kinds of experiments. (However, other experiments of different kind already refuted Abraham's and Bucherer's formulas long before.)

Poincaré stresses and the 4⁄3 problem

The idea of an electromagnetic nature of matter, however, had to be given up. Abraham (1904, 1905) argued that non-electromagnetic forces were necessary to prevent Lorentz's contractile electrons from exploding. He also showed that different results for the longitudinal electromagnetic mass can be obtained in Lorentz's theory, depending on whether the mass is calculated from its energy or its momentum, so a non-electromagnetic potential (corresponding to 1⁄3 of the electron's electromagnetic energy) was necessary to render these masses equal. Abraham doubted whether it was possible to develop a model satisfying all of these properties.

To solve those problems, Henri Poincaré in 1905 and 1906 introduced some sort of pressure ("Poincaré stresses") of non-electromagnetic nature. As required by Abraham, these stresses contribute non-electromagnetic energy to the electrons, amounting to 1⁄4 of their total energy or to 1⁄3 of their electromagnetic energy. So, the Poincaré stresses remove the contradiction in the derivation of the longitudinal electromagnetic mass, they prevent the electron from exploding, they remain unaltered by a Lorentz transformation (i.e. they are Lorentz invariant), and were also thought as a dynamical explanation of length contraction. However, Poincaré still assumed that only the electromagnetic energy contributes to the mass of the bodies.

As it was later noted, the problem lies in the 4⁄3 factor of electromagnetic rest mass – given above as ${\displaystyle m_{\mathrm {em} }={\tfrac {4}{3}}E_{\mathrm {em} }/c^{2}}$ when derived from the Abraham–Lorentz equations. However, when it is derived from the electron's electrostatic energy alone, we have ${\displaystyle m_{\mathrm {es} }=E_{\mathrm {em} }/c^{2}}$ where the 4⁄3 factor is missing. This can be solved by adding the non-electromagnetic energy ${\displaystyle E_{\mathrm {p} }}$ of the Poincaré stresses to ${\displaystyle E_{\mathrm {em} }}$, the electron's total energy ${\displaystyle E_{\mathrm {tot} }}$ now becomes:

${\displaystyle {\frac {E_{\mathrm {tot} }}{c^{2}}}={\frac {E_{\mathrm {em} }+E_{\mathrm {p} }}{c^{2}}}={\frac {E_{\mathrm {em} }+{\frac {E_{\mathrm {em} }}{3}}}{c^{2}}}={\frac {4}{3}}{\frac {E_{\mathrm {em} }}{c^{2}}}={\frac {4}{3}}m_{\mathrm {es} }=m_{\mathrm {em} }}$

Thus the missing 4⁄3 factor is restored when the mass is related to its electromagnetic energy, and it disappears when the total energy is considered.

Inertia of energy and radiation paradoxes

Radiation pressure

Another way of deriving some sort of electromagnetic mass was based on the concept of radiation pressure. These pressures or tensions in the electromagnetic field were derived by James Clerk Maxwell (1874) and Adolfo Bartoli (1876). Lorentz recognized in 1895 that those tensions also arise in his theory of the stationary aether. So if the electromagnetic field of the aether is able to set bodies in motion, the action / reaction principle demands that the aether must be set in motion by matter as well. However, Lorentz pointed out that any tension in the aether requires the mobility of the aether parts, which is not possible since in his theory the aether is immobile. (unlike contemporaries like Thomson who used fluid descriptions) This represents a violation of the reaction principle that was accepted by Lorentz consciously. He continued by saying, that one can only speak about fictitious tensions, since they are only mathematical models in his theory to ease the description of the electrodynamic interactions.

Mass of the fictitious electromagnetic fluid

In 1900 Poincaré studied the conflict between the action/reaction principle and Lorentz's theory. He tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields and radiation are involved. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum (such a momentum was also derived by Thomson in 1893 in a more complicated way). Poincaré concluded, the electromagnetic field energy behaves like a fictitious fluid („fluide fictif“) with a mass density of ${\displaystyle E_{em}/c^{2}}$ (in other words ${\displaystyle m_{em}=E_{em}/c^{2}}$). Now, if the center of mass frame (COM-frame) is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible – it is neither created or destroyed – then the motion of the center of mass frame remains uniform.

But this electromagnetic fluid is not indestructible, because it can be absorbed by matter (which according to Poincaré was the reason why he regarded the em-fluid as "fictitious" rather than "real"). Thus the COM-principle would be violated again. As it was later done by Einstein, an easy solution of this would be to assume that the mass of em-field is transferred to matter in the absorption process. But Poincaré created another solution: He assumed that there exists an immobile non-electromagnetic energy fluid at each point in space, also carrying a mass proportional to its energy. When the fictitious em-fluid is destroyed or absorbed, its electromagnetic energy and mass is not carried away by moving matter, but is transferred into the non-electromagnetic fluid and remains at exactly the same place in that fluid. (Poincaré added that one should not be too surprised by these assumptions, since they are only mathematical fictions.) In this way, the motion of the COM-frame, including matter, fictitious em-fluid, and fictitious non-em-fluid, at least theoretically remains uniform.

However, since only matter and electromagnetic energy are directly observable by experiment (not the non-em-fluid), Poincaré's resolution still violates the reaction principle and the COM-theorem, when an emission/absorption process is practically considered. This leads to a paradox when changing frames: if waves are radiated in a certain direction, the device will suffer a recoil from the momentum of the fictitious fluid. Then, Poincaré performed a Lorentz boost (to first order in v/c) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow perpetual motion, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether.

Poincaré came back to this topic in 1904. This time he rejected his own solution that motions in the ether can compensate the motion of matter, because any such motion is unobservable and therefore scientifically worthless. He also abandoned the concept that energy carries mass and wrote in connection to the above-mentioned recoil:

The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy.

These iterative developments culminated in his 1906 publication "The End of Matter" in which he notes that when applying the methodology of using an electric or magnetic field deviations to determine charge-to-mass ratios, one finds that the apparent mass added by charge makes up all of the apparent mass, thus the "real mass is equal to zero." Thus he goes on to postulate that electrons are only holes or motion effects in the aether while the aether itself is the only thing "endowed with inertia."

He then goes on to address the possibility that all matter might share this same quality and thereby his position changes from viewing aether as a "fictitious fluid" to suggesting it might be the only thing that actually exists in the universe, finally stating "In this system there is no actual matter, there are only holes in the aether."

Momentum and cavity radiation

However, Poincaré's idea of momentum and mass associated with radiation proved to be fruitful, when in 1903 Max Abraham introduced the term „electromagnetic momentum“, having a field density of ${\displaystyle E_{em}/c^{2}}$ per cm³ and ${\displaystyle E_{em}/c}$ per cm². Contrary to Lorentz and Poincaré, who considered momentum as a fictitious force, he argued that it is a real physical entity, and therefore conservation of momentum is guaranteed.

In 1904, Friedrich Hasenöhrl specifically associated inertia with radiation by studying the dynamics of a moving cavity. Hasenöhrl suggested that part of the mass of a body (which he called apparent mass) can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that ${\displaystyle m={\tfrac {8}{3}}E/c^{2}}$. However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation is not correct, and on the basis of his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to ${\displaystyle m={\tfrac {4}{3}}E/c^{2}}$, the same value for the electromagnetic mass for a body at rest. Hasenöhrl recalculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass that Poincaré would comment on in 1906. However, Hasenöhrl stated that this energy-apparent-mass relation only holds as long a body radiates, i.e. if the temperature of a body is greater than 0 K.

Modern view

Mass–energy equivalence

The idea that the principal relations between mass, energy, momentum and velocity can only be considered on the basis of dynamical interactions of matter was superseded, when Albert Einstein found out in 1905 that considerations based on the special principle of relativity require that all forms of energy (not only electromagnetic) contribute to the mass of bodies (mass–energy equivalence). That is, the entire mass of a body is a measure of its energy content by ${\displaystyle E=mc^{2}}$, and Einstein's considerations were independent from assumptions about the constitution of matter. By this equivalence, Poincaré's radiation paradox can be solved without using "compensating forces", because the mass of matter itself (not the non-electromagnetic aether fluid as suggested by Poincaré) is increased or diminished by the mass of electromagnetic energy in the course of the emission/absorption process. Also the idea of an electromagnetic explanation of gravitation was superseded in the course of developing general relativity.

So every theory dealing with the mass of a body must be formulated in a relativistic way from the outset. This is for example the case in the current quantum field explanation of mass of elementary particles in the framework of the Standard Model, the Higgs mechanism. Because of this, the idea that any form of mass is completely caused by interactions with electromagnetic fields, is not relevant any more.

Relativistic mass

The concepts of longitudinal and transverse mass (equivalent to those of Lorentz) were also used by Einstein in his first papers on relativity. However, in special relativity they apply to the entire mass of matter, not only to the electromagnetic part. Later it was shown by physicists like Richard Chace Tolman that expressing mass as the ratio of force and acceleration is not advantageous. Therefore, a similar concept without direction dependent terms, in which force is defined as ${\displaystyle {\vec {F}}=\mathrm {d} {\vec {p}}/\mathrm {d} t}$, was used as relativistic mass

${\displaystyle M={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\qquad m_{0}={\frac {E}{c^{2}}},}$

This concept is sometimes still used in modern physics textbooks, although the term 'mass' is now considered by many to refer to invariant mass, see mass in special relativity.

Self-energy

When the special case of the electromagnetic self-energy or self-force of charged particles is discussed, also in modern texts some sort of "effective" electromagnetic mass is sometimes introduced – not as an explanation of mass per se, but in addition to the ordinary mass of bodies. Many different reformulations of the Abraham–Lorentz force have been derived – for instance, in order to deal with the 4⁄3-problem (see next section) and other problems that arose from this concept. Such questions are discussed in connection with renormalization, and on the basis of quantum mechanics and quantum field theory, which must be applied when the electron is considered physically point-like. At distances located in the classical domain, the classical concepts again come into play. A rigorous derivation of the electromagnetic self-force, including the contribution to the mass of the body, was published by Gralla et al. (2009).

4⁄3 problem

Max von Laue in 1911 also used the Abraham–Lorentz equations of motion in his development of special relativistic dynamics, so that also in special relativity the 4⁄3 factor is present when the electromagnetic mass of a charged sphere is calculated. This contradicts the mass–energy equivalence formula, which requires the relation ${\displaystyle m_{\mathrm {em} }=E_{\mathrm {em} }/c^{2}}$ without the 4⁄3 factor, or in other words, four-momentum doesn't properly transform like a four-vector when the 4⁄3 factor is present. Laue found a solution equivalent to Poincaré's introduction of a non-electromagnetic potential (Poincaré stresses), but Laue showed its deeper, relativistic meaning by employing and advancing Hermann Minkowski's spacetime formalism. Laue's formalism required that there are additional components and forces, which guarantee that spatially extended systems (where both electromagnetic and non-electromagnetic energies are combined) are forming a stable or "closed system" and transform as a four-vector. That is, the 4⁄3 factor arises only with respect to electromagnetic mass, while the closed system has total rest mass and energy of ${\displaystyle m_{\mathrm {tot} }=E_{\mathrm {tot} }/c^{2}}$.

Another solution was found by authors such as Enrico Fermi (1922), Paul Dirac (1938) Fritz Rohrlich (1960), or Julian Schwinger (1983), who pointed out that the electron's stability and the 4/3-problem are two different things. They showed that the preceding definitions of four-momentum are non-relativistic per se, and by changing the definition into a relativistic form, the electromagnetic mass can simply be written as ${\displaystyle m_{\mathrm {em} }=E_{\mathrm {em} }/c^{2}}$ and thus the 4⁄3 factor doesn't appear at all. So every part of the system, not only "closed" systems, properly transforms as a four-vector. However, binding forces like the Poincaré stresses are still necessary to prevent the electron from exploding due to Coulomb repulsion. But on the basis of the Fermi–Rohrlich definition, this is only a dynamical problem and has nothing to do with the transformation properties any more.

Also other solutions have been proposed, for instance, Valery Morozov (2011) gave consideration to movement of an imponderable charged sphere. It turned out that a flux of nonelectromagnetic energy exists in the sphere body. This flux has an impulse exactly equal to 1⁄3 of the sphere electromagnetic impulse regardless of a sphere internal structure or a material, it is made of. The problem was solved without attraction of any additional hypotheses. In this model, sphere tensions are not connected with its mass.

Higgs mechanism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Higgs_mechanism 

 

Standard Model of particle physics
Elementary particles of the Standard Model

In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W⁺, W⁻, and Z⁰ bosons actually have relatively large masses of around 80 GeV/c². The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all Hilbert spaces of the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W^±, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature.

The mechanism was proposed in 1962 by Philip Warren Anderson, following work in the late 1950s on symmetry breaking in superconductivity and a 1960 paper by Yoichiro Nambu that discussed its application within particle physics.

A theory able to finally explain mass generation without "breaking" gauge theory was published almost simultaneously by three independent groups in 1964: by Robert Brout and François Englert; by Peter Higgs; and by Gerald Guralnik, C. R. Hagen, and Tom Kibble. The Higgs mechanism is therefore also called the Brout–Englert–Higgs mechanism, or Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism, Anderson–Higgs mechanism, Anderson–Higgs–Kibble mechanism, Higgs–Kibble mechanism by Abdus Salam and ABEGHHK'tH mechanism (for Anderson, Brout, Englert, Guralnik, Hagen, Higgs, Kibble, and 't Hooft) by Peter Higgs. The Higgs mechanism in electrodynamics was also discovered independently by Eberly and Reiss in reverse as the "gauge" Dirac field mass gain due to the artificially displaced electromagnetic field as a Higgs field.

On 8 October 2013, following the discovery at CERN's Large Hadron Collider of a new particle that appeared to be the long-sought Higgs boson predicted by the theory, it was announced that Peter Higgs and François Englert had been awarded the 2013 Nobel Prize in Physics.

Standard Model

The Higgs mechanism was incorporated into modern particle physics by Steven Weinberg and Abdus Salam, and is an essential part of the Standard Model.

In the Standard Model, at temperatures high enough that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature, the Higgs field develops a vacuum expectation value; the symmetry is spontaneously broken by tachyon condensation, and the W and Z bosons acquire masses (also called "electroweak symmetry breaking", or EWSB). In the history of the universe, this is believed to have happened about a picosecond (10⁻¹² s) after the hot big bang, when the universe was at a temperature 159.5 ± 1.5 GeV.

Fermions, such as the leptons and quarks in the Standard Model, can also acquire mass as a result of their interaction with the Higgs field, but not in the same way as the gauge bosons.

Structure of the Higgs field

In the standard model, the Higgs field is an SU(2) doublet (i.e. the standard representation with two complex components called isospin), which is a scalar under Lorentz transformations. Its electric charge is zero; its weak isospin is 1/2 and the third component of weak isospin is −1/2; and its weak hypercharge (the charge for the U(1) gauge group defined up to an arbitrary multiplicative constant) is 1. Under U(1) rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other, combining to the standard two-component complex representation of the group U(2).

The Higgs field, through the interactions specified (summarized, represented, or even simulated) by its potential, induces spontaneous breaking of three out of the four generators ("directions") of the gauge group U(2). This is often written as SU(2)_L × U(1)_Y, (which is strictly speaking only the same on the level of infinitesimal symmetries) because the diagonal phase factor also acts on other fields – quarks in particular. Three out of its four components would ordinarily resolve as Goldstone bosons, if they were not coupled to gauge fields.

However, after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three W and Z bosons (
W⁺
,
W⁻
and
Z⁰
), and are only observable as components of these weak bosons, which are made massive by their inclusion; only the single remaining degree of freedom becomes a new scalar particle: the Higgs boson. The components that do not mix with Goldstone bosons form a massless photon.

The photon as the part that remains massless

The gauge group of the electroweak part of the standard model is SU(2)_L × U(1)_Y. The group SU(2) is the group of all 2-by-2 unitary matrices with unit determinant; all the orthonormal changes of coordinates in a complex two dimensional vector space.

Rotating the coordinates so that the second basis vector points in the direction of the Higgs boson makes the vacuum expectation value of H the spinor (0, v). The generators for rotations about the x, y, and z axes are by half the Pauli matrices σ_x, σ_y, and σ_z, so that a rotation of angle θ about the z-axis takes the vacuum to

${\displaystyle {\bigl (}0,ve^{-{\frac {1}{2}}i\theta }{\bigr )}.}$

While the T_x and T_y generators mix up the top and bottom components of the spinor, the T_z rotations only multiply each by opposite phases. This phase can be undone by a U(1) rotation of angle 1/2θ. Consequently, under both an SU(2) T_z-rotation and a U(1) rotation by an amount 1/2θ, the vacuum is invariant.

This combination of generators

${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y}$

defines the unbroken part of the gauge group, where Q is the electric charge, T₃ is the generator of rotations around the 3-axis in the SU(2) and Y is the hypercharge generator of the U(1). This combination of generators (a 3 rotation in the SU(2) and a simultaneous U(1) rotation by half the angle) preserves the vacuum, and defines the unbroken gauge group in the standard model, namely the electric charge group. The part of the gauge field in this direction stays massless, and amounts to the physical photon.

Consequences for fermions

In spite of the introduction of spontaneous symmetry breaking, the mass terms preclude chiral gauge invariance. For these fields, the mass terms should always be replaced by a gauge-invariant "Higgs" mechanism. One possibility is some kind of Yukawa coupling (see below) between the fermion field ψ and the Higgs field Φ, with unknown couplings G_ψ, which after symmetry breaking (more precisely: after expansion of the Lagrange density around a suitable ground state) again results in the original mass terms, which are now, however (i.e., by introduction of the Higgs field) written in a gauge-invariant way. The Lagrange density for the Yukawa interaction of a fermion field ψ and the Higgs field Φ is

${\displaystyle {\mathcal {L}}_{\mathrm {Fermion} }(\phi ,A,\psi )={\overline {\psi }}\gamma ^{\mu }D_{\mu }\psi +G_{\psi }{\overline {\psi }}\phi \psi ,}$

where again the gauge field A only enters via the gauge covariant derivative operator D_μ (i.e., it is only indirectly visible). The quantities γ^μ are the Dirac matrices, and G_ψ is the already-mentioned Yukawa coupling parameter. Now the mass-generation follows the same principle as above, namely from the existence of a finite expectation value ${\displaystyle |\langle \phi \rangle |}$. Again, this is crucial for the existence of the property mass.

History of research

Background

Spontaneous symmetry breaking offered a framework to introduce bosons into relativistic quantum field theories. However, according to Goldstone's theorem, these bosons should be massless. The only observed particles which could be approximately interpreted as Goldstone bosons were the pions, which Yoichiro Nambu related to chiral symmetry breaking.

A similar problem arises with Yang–Mills theory (also known as non-abelian gauge theory), which predicts massless spin-1 gauge bosons. Massless weakly-interacting gauge bosons lead to long-range forces, which are only observed for electromagnetism and the corresponding massless photon. Gauge theories of the weak force needed a way to describe massive gauge bosons in order to be consistent.

Discovery

Philip W. Anderson, the first to implement the mechanism in 1962.

 

Five of the six 2010 APS Sakurai Prize Winners – (L to R) Tom Kibble, Gerald Guralnik, Carl Richard Hagen, François Englert, and Robert Brout

 

Peter Higgs (2009)

That breaking gauge symmetries did not lead to massless particles was observed in 1961 by Julian Schwinger, but he did not demonstrate massive particles would eventuate. This was done in Philip Warren Anderson's 1962 paper but only in non-relativistic field theory; it also discussed consequences for particle physics but did not work out an explicit relativistic model. The relativistic model was developed in 1964 by three independent groups:

• Robert Brout and François Englert
• Peter Higgs
• Gerald Guralnik, Carl Richard Hagen, and Tom Kibble.

Slightly later, in 1965, but independently from the other publications the mechanism was also proposed by Alexander Migdal and Alexander Polyakov, at that time Soviet undergraduate students. However, their paper was delayed by the editorial office of JETP, and was published late, in 1966.

The mechanism is closely analogous to phenomena previously discovered by Yoichiro Nambu involving the "vacuum structure" of quantum fields in superconductivity. A similar but distinct effect (involving an affine realization of what is now recognized as the Higgs field), known as the Stueckelberg mechanism, had previously been studied by Ernst Stueckelberg.

These physicists discovered that when a gauge theory is combined with an additional field that spontaneously breaks the symmetry group, the gauge bosons can consistently acquire a nonzero mass. In spite of the large values involved (see below) this permits a gauge theory description of the weak force, which was independently developed by Steven Weinberg and Abdus Salam in 1967. Higgs's original article presenting the model was rejected by Physics Letters. When revising the article before resubmitting it to Physical Review Letters, he added a sentence at the end, mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete representations of the symmetry group; these are the Higgs bosons.

The three papers by Brout and Englert; Higgs; and Guralnik, Hagen, and Kibble were each recognized as "milestone letters" by Physical Review Letters in 2008. While each of these seminal papers took similar approaches, the contributions and differences among the 1964 PRL symmetry breaking papers are noteworthy. All six physicists were jointly awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics for this work.

Benjamin W. Lee is often credited with first naming the "Higgs-like" mechanism, although there is debate around when this first occurred. One of the first times the Higgs name appeared in print was in 1972 when Gerardus 't Hooft and Martinus J. G. Veltman referred to it as the "Higgs–Kibble mechanism" in their Nobel winning paper.

Simple explanation of the theory, from its origins in superconductivity

The proposed Higgs mechanism arose as a result of theories proposed to explain observations in superconductivity. A superconductor does not allow penetration by external magnetic fields (the Meissner effect). This strange observation implies that somehow, the electromagnetic field becomes short ranged during this phenomenon. Successful theories arose to explain this during the 1950s, first for fermions (Ginzburg–Landau theory, 1950), and then for bosons (BCS theory, 1957).

In these theories, superconductivity is interpreted as arising from a charged condensate. Initially, the condensate value does not have any preferred direction, implying it is scalar, but its phase is capable of defining a gauge, in gauge based field theories. To do this, the field must be charged. A charged scalar field must also be complex (or described another way, it contains at least two components, and a symmetry capable of rotating each into the other(s)). In naïve gauge theory, a gauge transformation of a condensate usually rotates the phase. But in these circumstances, it instead fixes a preferred choice of phase. However it turns out that fixing the choice of gauge so that the condensate has the same phase everywhere, also causes the electromagnetic field to gain an extra term. This extra term causes the electromagnetic field to become short range.

(Goldstone's theorem also plays a role in such theories. The connection is technically, when a condensate breaks a symmetry, then the state reached by acting with a symmetry generator on the condensate has the same energy as before. This means that some kinds of oscillation will not involve change of energy. Oscillations with unchanged energy imply that excitations (particles) associated with the oscillation are massless.)

Once attention was drawn to this theory within particle physics, the parallels were clear. A change of the usually long range electromagnetic field to become short ranged, within a gauge invariant theory, was exactly the needed effect sought for the weak force bosons (because a long range force has massless gauge bosons, and a short ranged force implies massive gauge bosons, suggesting that a result of this interaction is that the field's gauge bosons acquired mass, or a similar and equivalent effect). The features of a field required to do this was also quite well defined - it would have to be a charged scalar field, with at least two components, and complex in order to support a symmetry able to rotate these into each other.

Examples

The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. In the non-relativistic context this is a superconductor, more formally known as the Landau model of a charged Bose–Einstein condensate. In the relativistic condensate, the condensate is a scalar field that is relativistically invariant.

Landau model

The Higgs mechanism is a type of superconductivity which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged, or, in field language, when a charged field has a nonzero vacuum expectation value. Interaction with the quantum fluid filling the space prevents certain forces from propagating over long distances (as it does inside a superconductor; e.g., in the Ginzburg–Landau theory).

A superconductor expels all magnetic fields from its interior, a phenomenon known as the Meissner effect. This was mysterious for a long time, because it implies that electromagnetic forces somehow become short-range inside the superconductor. Contrast this with the behavior of an ordinary metal. In a metal, the conductivity shields electric fields by rearranging charges on the surface until the total field cancels in the interior.

But magnetic fields can penetrate to any distance, and if a magnetic monopole (an isolated magnetic pole) is surrounded by a metal the field can escape without collimating into a string. In a superconductor, however, electric charges move with no dissipation, and this allows for permanent surface currents, not just surface charges. When magnetic fields are introduced at the boundary of a superconductor, they produce surface currents which exactly neutralize them.

The Meissner effect arises due to currents in a thin surface layer, whose thickness can be calculated from the simple model of Ginzburg–Landau theory, which treats superconductivity as a charged Bose–Einstein condensate.

Suppose that a superconductor contains bosons with charge q. The wavefunction of the bosons can be described by introducing a quantum field, ψ, which obeys the Schrödinger equation as a field equation. In units where the reduced Planck constant, ħ, is set to 1:

${\displaystyle i{\partial \over \partial t}\psi ={(\nabla -iqA)^{2} \over 2m}\psi .}$

The operator ψ(x) annihilates a boson at the point x, while its adjoint ψ^† creates a new boson at the same point. The wavefunction of the Bose–Einstein condensate is then the expectation value ψ of ψ(x), which is a classical function that obeys the same equation. The interpretation of the expectation value is that it is the phase that one should give to a newly created boson so that it will coherently superpose with all the other bosons already in the condensate.

When there is a charged condensate, the electromagnetic interactions are screened. To see this, consider the effect of a gauge transformation on the field. A gauge transformation rotates the phase of the condensate by an amount which changes from point to point, and shifts the vector potential by a gradient:

${\displaystyle {\begin{aligned}\psi &\rightarrow e^{iq\phi (x)}\psi \\A&\rightarrow A+\nabla \phi .\end{aligned}}}$

When there is no condensate, this transformation only changes the definition of the phase of ψ at every point. But when there is a condensate, the phase of the condensate defines a preferred choice of phase.

The condensate wave function can be written as

${\displaystyle \psi (x)=\rho (x)\,e^{i\theta (x)},}$

where ρ is real amplitude, which determines the local density of the condensate. If the condensate were neutral, the flow would be along the gradients of θ, the direction in which the phase of the Schrödinger field changes. If the phase θ changes slowly, the flow is slow and has very little energy. But now θ can be made equal to zero just by making a gauge transformation to rotate the phase of the field.

The energy of slow changes of phase can be calculated from the Schrödinger kinetic energy,

${\displaystyle H={1 \over 2m}\left|(iqA+\nabla )\psi \right|^{2},}$

and taking the density of the condensate ρ to be constant,

${\displaystyle H\approx {\rho ^{2} \over 2m}(qA+\nabla \theta )^{2}.}$

Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term,

${\displaystyle {q^{2}\rho ^{2} \over 2m}A^{2}.}$

When this term is present, electromagnetic interactions become short-ranged. Every field mode, no matter how long the wavelength, oscillates with a nonzero frequency. The lowest frequency can be read off from the energy of a long wavelength A mode,

${\displaystyle E\approx {{\dot {A}}^{2} \over 2}+{q^{2}\rho ^{2} \over 2m}A^{2}.}$

This is a harmonic oscillator with frequency

${\displaystyle {\sqrt {{\frac {1}{m}}q^{2}\rho ^{2}}}.}$

The quantity |ψ|² (= ρ²) is the density of the condensate of superconducting particles.

In an actual superconductor, the charged particles are electrons, which are fermions not bosons. So in order to have superconductivity, the electrons need to somehow bind into Cooper pairs. The charge of the condensate q is therefore twice the electron charge −e. The pairing in a normal superconductor is due to lattice vibrations, and is in fact very weak; this means that the pairs are very loosely bound. The description of a Bose–Einstein condensate of loosely bound pairs is actually more difficult than the description of a condensate of elementary particles, and was only worked out in 1957 by John Bardeen, Leon Cooper and John Robert Schrieffer in the famous BCS theory.

Abelian Higgs mechanism

Gauge invariance means that certain transformations of the gauge field do not change the energy at all. If an arbitrary gradient is added to A, the energy of the field is exactly the same. This makes it difficult to add a mass term, because a mass term tends to push the field toward the value zero. But the zero value of the vector potential is not a gauge invariant idea. What is zero in one gauge is nonzero in another.

So in order to give mass to a gauge theory, the gauge invariance must be broken by a condensate. The condensate will then define a preferred phase, and the phase of the condensate will define the zero value of the field in a gauge-invariant way. The gauge-invariant definition is that a gauge field is zero when the phase change along any path from parallel transport is equal to the phase difference in the condensate wavefunction.

The condensate value is described by a quantum field with an expectation value, just as in the Ginzburg–Landau model.

In order for the phase of the vacuum to define a gauge, the field must have a phase (also referred to as 'to be charged'). In order for a scalar field Φ to have a phase, it must be complex, or (equivalently) it should contain two fields with a symmetry which rotates them into each other. The vector potential changes the phase of the quanta produced by the field when they move from point to point. In terms of fields, it defines how much to rotate the real and imaginary parts of the fields into each other when comparing field values at nearby points.

The only renormalizable model where a complex scalar field Φ acquires a nonzero value is the Mexican-hat model, where the field energy has a minimum away from zero. The action for this model is

${\displaystyle S(\phi )=\int d^{4}x\left[{\frac {1}{2}}\left|\partial \phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right],}$

which results in the Hamiltonian

${\displaystyle H(\phi )={\frac {1}{2}}\left(\left|{\dot {\phi }}\right|^{2}+\left|\nabla \phi \right|^{2}\right)+V(\left|\phi \right|).}$

The first term is the kinetic energy of the field. The second term is the extra potential energy when the field varies from point to point. The third term is the potential energy when the field has any given magnitude.

This potential energy, the Higgs potential, z, has a graph which looks like a Mexican hat, which gives the model its name. In particular, the minimum energy value is not at z = 0, but on the circle of points where the magnitude of z is Φ.

Higgs potential V. For a fixed value of λ, the potential is presented upwards against the real and imaginary parts of Φ. The Mexican-hat or champagne-bottle profile at the ground should be noted.

When the field Φ(x) is not coupled to electromagnetism, the Mexican-hat potential has flat directions. Starting in any one of the circle of vacua and changing the phase of the field from point to point costs very little energy. Mathematically, if

${\displaystyle \phi (x)=\Phi e^{i\theta (x)}}$

with a constant prefactor, then the action for the field θ(x), i.e., the "phase" of the Higgs field Φ(x), has only derivative terms. This is not a surprise. Adding a constant to θ(x) is a symmetry of the original theory, so different values of θ(x) cannot have different energies. This is an example of Goldstone's theorem: spontaneously broken continuous symmetries normally produce massless excitations.

The Abelian Higgs model is the Mexican-hat model coupled to electromagnetism:

${\displaystyle S(\phi ,A)=\int d^{4}x\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+\left|\left(\partial -iqA\right)\phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right].}$

Alternate form of the Abelian Higgs model action

The classical vacuum is again at the minimum of the potential, where the magnitude of the complex field φ is equal to Φ. But now the phase of the field is arbitrary, because gauge transformations change it. This means that the field ${\displaystyle \theta (x)}$ can be set to zero by a gauge transformation, and does not represent any actual degrees of freedom at all.

Furthermore, choosing a gauge where the phase of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero. So in the Abelian Higgs model, the gauge field acquires a mass. To calculate the magnitude of the mass, consider a constant value of the vector potential A in the x-direction in the gauge where the condensate has constant phase. This is the same as a sinusoidally varying condensate in the gauge where the vector potential is zero. In the gauge where A is zero, the potential energy density in the condensate is the scalar gradient energy:

${\displaystyle E={\frac {1}{2}}\left|\partial \left(\Phi e^{iqAx}\right)\right|^{2}={\tfrac {1}{2}}q^{2}\Phi ^{2}A^{2}.}$

This energy is the same as a mass term 1/2m²A² where m = q Φ.

Mathematical details of the abelian Higgs mechanism

Lagrangian in explicit symmetry broken form

Spontaneous symmetry breaking and trivializations

Non-Abelian Higgs mechanism

The Non-Abelian Higgs model has the following action

${\displaystyle S(\phi ,\mathbf {A} )=\int {1 \over 4g^{2}}\mathop {\textrm {tr}} (F^{\mu \nu }F_{\mu \nu })+|D\phi |^{2}+V(|\phi |),}$

where now the non-Abelian field A is contained in the covariant derivative D and in the tensor components ${\displaystyle F^{\mu \nu }}$ and ${\displaystyle F_{\mu \nu }}$ (the relation between A and those components is well-known from the Yang–Mills theory).

It is exactly analogous to the Abelian Higgs model. Now the field ${\displaystyle \phi }$ is in a representation of the gauge group, and the gauge covariant derivative is defined by the rate of change of the field minus the rate of change from parallel transport using the gauge field A as a connection.

${\displaystyle D\phi =\partial \phi -iA^{k}t_{k}\phi }$

Again, the expectation value of ${\displaystyle \phi }$ defines a preferred gauge where the vacuum is constant, and fixing this gauge, fluctuations in the gauge field A come with a nonzero energy cost.

Depending on the representation of the scalar field, not every gauge field acquires a mass. A simple example is in the renormalizable version of an early electroweak model due to Julian Schwinger. In this model, the gauge group is SO(3) (or SU(2) − there are no spinor representations in the model), and the gauge invariance is broken down to U(1) or SO(2) at long distances. To make a consistent renormalizable version using the Higgs mechanism, introduce a scalar field ${\displaystyle \phi ^{a}}$ which transforms as a vector (a triplet) of SO(3). If this field has a vacuum expectation value, it points in some direction in field space. Without loss of generality, one can choose the z-axis in field space to be the direction that ${\displaystyle \phi }$ is pointing, and then the vacuum expectation value of ${\displaystyle \phi }$ is (0, 0, Ã), where à is a constant with dimensions of mass (${\displaystyle c=\hbar =1}$).

Rotations around the z-axis form a U(1) subgroup of SO(3) which preserves the vacuum expectation value of ${\displaystyle \phi }$, and this is the unbroken gauge group. Rotations around the x and y-axis do not preserve the vacuum, and the components of the SO(3) gauge field which generate these rotations become massive vector mesons. There are two massive W mesons in the Schwinger model, with a mass set by the mass scale Ã, and one massless U(1) gauge boson, similar to the photon.

The Schwinger model predicts magnetic monopoles at the electroweak unification scale, and does not predict the Z boson. It doesn't break electroweak symmetry properly as in nature. But historically, a model similar to this (but not using the Higgs mechanism) was the first in which the weak force and the electromagnetic force were unified.

Affine Higgs mechanism

Ernst Stueckelberg discovered a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon. Effectively, Stueckelberg's model is a limit of the regular Mexican hat Abelian Higgs model, where the vacuum expectation value H goes to infinity and the charge of the Higgs field goes to zero in such a way that their product stays fixed. The mass of the Higgs boson is proportional to H, so the Higgs boson becomes infinitely massive and decouples, so is not present in the discussion. The vector meson mass, however, is equal to the product eH, and stays finite.

The interpretation is that when a U(1) gauge field does not require quantized charges, it is possible to keep only the angular part of the Higgs oscillations, and discard the radial part. The angular part of the Higgs field θ has the following gauge transformation law:

${\displaystyle {\begin{aligned}\theta &\rightarrow \theta +e\alpha \,\\A&\rightarrow A+\partial \alpha .\end{aligned}}}$

The gauge covariant derivative for the angle (which is actually gauge invariant) is:

${\displaystyle D\theta =\partial \theta -eAH\,}$.

In order to keep θ fluctuations finite and nonzero in this limit, θ should be rescaled by H, so that its kinetic term in the action stays normalized. The action for the theta field is read off from the Mexican hat action by substituting ${\displaystyle \phi =He^{i\theta /H}}$.

${\displaystyle S=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(D\theta )^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -HeA)^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -mA)^{2}{\bigr ]}}$

since eH is the gauge boson mass. By making a gauge transformation to set θ = 0, the gauge freedom in the action is eliminated, and the action becomes that of a massive vector field:

${\displaystyle S={\tfrac {1}{2}}\int {\bigl [}{\tfrac {1}{2}}F^{2}+m^{2}A^{2}{\bigr ]}\,.}$

To have arbitrarily small charges requires that the U(1) is not the circle of unit complex numbers under multiplication, but the real numbers R under addition, which is only different in the global topology. Such a U(1) group is non-compact. The field θ transforms as an affine representation of the gauge group. Among the allowed gauge groups, only non-compact U(1) admits affine representations, and the U(1) of electromagnetism is experimentally known to be compact, since charge quantization holds to extremely high accuracy.

The Higgs condensate in this model has infinitesimal charge, so interactions with the Higgs boson do not violate charge conservation. The theory of quantum electrodynamics with a massive photon is still a renormalizable theory, one in which electric charge is still conserved, but magnetic monopoles are not allowed. For non-Abelian gauge theory, there is no affine limit, and the Higgs oscillations cannot be too much more massive than the vectors.