Are electrons pointlike/structureless?

Elementary particles are considered to be point-like, but not point particles. QED, or relativistic quantum field theory in general, is _not_ based on the notion of ''point particles'', as one sees stated so often and yet so erroneously. (emphasis as in the original)

This quote from p.2 of the book

tells everything. But why is this so?

A point particle is the idealization of a real particle seen from so far away that scattering of other particles is as if the given particle were a point. Specifically, a relativistic charged particle is considered to be a point particle at the energies of interest if its interaction with an external electromagnetic field can be accurately described by the Dirac equation.


Both electrons and neutrinos are considered to be pointlike because of the way they appear in the standard model. Pointlike means that the associated bare particles are points. But these bare particles are very strange objects. According to renormalization theory, the basis of modern QED and other relativistic field theories, bare electrons have no associated electromagnetic field although they have an infinite charge (and an infinite mass) -- something inconsistent with real physics. They do not exist.

The bare particles are points = structureless formal building blocks of the theories with which (after renormalization = dressing) the physical = real = dressed = observable particles are described. The latter have a nontrivial electromagnetic structure encoded in their form factors. (The term ''dresses'' comes from an intuitive picture form the early days of quantum field theory, where a dressed particle was viewed as the corresponding bare particle dressed in a shirt made of infinitely many soft bare photons and bareparticle-antiparticle pairs.)

Physical, measurable particles are not points but have extension. By definition, an electron without extension would be described exactly by the 1-particle Dirac equation, which has a degenerate spectrum. But the real electron is described by a modified Dirac equation, in which the so-called form factors figure. These are computable from QED, resulting in an anomalous magnetic moment and a nonzero Lamb shift removing the degeneracy of the spectrum. Both are measurable to high accuracy, and are not present for point particles, which by definition satisfy the Dirac equation exactly.


The size of a particle is determined by how the particle responds to scattering experiments, and therefore is (like the size of a balloon) somewhat context-dependent. (The context is given by a wave function and determines the detailed state of the particle.)

On the other hand, the deviations from being a point are usually described by means of context-independent form factors that would be constant for a point particle but become momentum-dependent for particles in general. They characterize the particle as a state-independent entity. Together with a particle's state, the form factors contain everything that can be observed about single particles in an electromagnetic field.

An electron has two form factors, a magnetic and an electric one.

The electric form factor determines the charge radius of a particle, defined as the number r such that the electric form factor has an expansion of the form (Units are such that c=1 and hbar=1.) This definition is motivated by the fact that the average over exp(i q dot x) over a spherical shell of radius r has this asymptotic behavior. See Formula (11.3.32) in QED, which treats the electron as pointlike in the usual sense of the word - that it appears as a fundamental field in the Lagrangian -, implies a positive value for the charge radius of the electron. Indeed, this is Weinberg's conclusion from his calculations in Section 11.3, together with an estimate of infrared effects taken from (14.3.1).


The mathematically most accessible derivation of the electron form factor is in the book

Section 3.9 starts off with: It is often said that the electron is a point particle without structure in contrast to the proton, for example. We will see in this section that this is not true. The electromagnetic structure of the electron is contained in the form factors.


The relations between form factors for spin 1/2 particles and terms in a modified Dirac equation describing the covariant dynamics in an electromagnetic field of a particle deviating from a point particle are given in

According to Foldy, electromagnetic form factors of an elementary or compound particle (in Foldy's 1952 paper described by a modified Dirac equation, so it applies to the electron and the nucleon, but bosons and nonrelativistic particles can be handled similarly) characterize the charge and current distribution of the particle (in the nonrelativistic case simply via a Fourier transform) and its response to an external electromagnetic field.

The anomalous magnetic moment shows directly as a coefficient in the modified Dirac equation. On the other hand, the spectral resolution of the single-particle problem in a constant electric field leads to a spectrum from which the Lamb shift can be read off.

Foldy's paper gives a very clear analysis of the physical interpretation of the form factors, though he does not use this terminology for his coefficients. But this connection is drawn in the paper

Both papers don't address the changes of the dynamics due to the nontrivial electron self-energy. This leads to additional corrections of the kinetic (i.e., derivative) term in the Dirac equation, while the form factors only modify the (nonderivative) interaction. (See the FAQ entry on self-energy.)


To probe the electric form factors, one usually uses scattering experiments and fits their results to phenomenological expressions for the form factor. When scattering electrons off the particle, the electrons respond to the form factors of the particle, in a nonrelativistic treatment more precisely to the charge distribution which can then be computed from the electric form factor:

The relativistic cross section involves both form factors; see (2.14) in Yennie et al..

An intuitive argument for the extendedness of relativistic particles (which is difficult to make precise, though) is the fact that their localization to a region significantly smaller than the de Broglie wavelength would need energies larger than that needed to create particle-antiparticle pairs, which changes the nature of the system. Note, however, that even point particles (which satisfy by definition the Dirac equation exactly) have a nonzero Compton wavelength. (Localization is discussed elsewhere in this FAQ; see also Foldy's papers quoted there.)

The terminology ''particle size'' is not used in a well-defined way. But there are precise related concepts; e.g., the charge radius. it is positive even for neutral particles such as the neutron or the neutrino, but is zero for a Dirac particle. (The latter is a theoretical entity only.)


On a more formal, quantitative level, the physical, dressed particles have nontrivial form factors, due to the renormalization necessary to give finite results in QFT. The form factor measures the deviation from the behavior of an ideal point particle, i.e., a particle obeying exactly the the Dirac equation. The form factor of the electron can be computed perturbatively from QED, and it can be indirectly determined by experiment, e.g., through the observation of the anomalous magnetic moment and the Lamb shift. (A point particle has no anomalous magnetic moment and no Lamb shift since it satisfies the Dirac equation exactly.)


The "form" of an elementary particle (considered as a free particle at rest) is described by its form factor, which is a well-defined physical function (though at present computable only in perturbation theory) describing how the (spin 0, 1/2, or 1) particle's response to an external classical electromagnetic field deviates from the Klein-Gordon, Dirac, or Maxwell equations, respectively.

The form factor contains the complete state-independent information about a free particle, since it determines the (single-particle) Hamilton operator of the free particle and everything else can be computed from it. (But it is an approximate description only since it ignores memory effects that would arise in an exact treatment of the dynamics.)

The paper

discusses the extendedness of the electron in a phenomenological way. In Foldy's paper, the form factors are encoded in the infinite sum in (16). The sum is usually considered in the momentum domain; then one simply gets two k-dependent form factors, where k represents the 4-momentum transferred in the interaction. These form factors can be calculated in a good approximation perturbatively from QFT, see for example Peskin and Schroeder's book.

An extensive discussion of form factors of Dirac particles and their relation to the radial density function is in the paper by Yennie et al. cited above, and in

Yennie et al. write: Information about the internal structure of the individual nucleons is contained in the results of a variety of experiments performed in recent years. [...] The Lamb shift and the hyperfine splitting also give such information, [...] The charge-current density of the nucleon (proton or neutron) includes all of the effects of the internal structure. [...] The nucleon charge-current density must have the form The functions F_1 and F_2 are relativistic generalizations of the form factors characteristic of finite extension occurring in other experiments, [...]


Nontrivial form factors give rise to a positive charge radius. In his book cited above, Weinberg explicitly computes in (11.3.33) a formula for the charge radius of a physical electron. But his formula is not fully satisfying since it suffers from an infrared divergence: the expression contains a fictitious photon mass, and diverges if this goes to zero, as infrared corrections from soft photons are not accounted for). See also Section V in the review article

where the authors say: According to QED an electron continuously emits and absorbs virtual photons (see the leading order diagram in Fig. 8) and as a result its electric charge is spread over a finite volume instead of being pointlike. Then they give without proof the explicit formula (28) for the charge radius, depending logarithmically on the charge of the central field in which the electron moves. Thus, because of the binding-energy-dependent cutoff used to regularize the divergence, the electron charge radius depends on its surrounding. (Indeed, as known from the general mechanism of deriving approximate reduced descriptions of a subsystem of a bigger system, the cutoff is always determined by the environment, and is independent of the state of the subsystem.) Thus the electron appears to be a compressible substance.


Of course, other particles also have form factors and associated charge radii. For proton and neutron form factors, see hep-ph/0204239 and hep-ph/030305. Neutrons have a negative mean squared charge radius. This looks strange but is not since the measure for the mean is not positive; but it means that a classical interpretation of the charge radius of neutrons is dubious. In the introduction of

one can read: The charge radius of the neutron <r_n^2> or the mean squared charge radius is described by the volume integral over the neutron integral rho(r)r^2dr, where r is the distance to the center of the neutron and rho(r) is the charge density. Positive as well as negative values of rho(r) will occur coming from the distributions of valence quarks and the negative p-meson cloud outside. Since rho(r) is negative for larger r values, caused by the meson cloud, the r^2 dependence of the integral will lead to a negative value of <r_n^2>.


On the numerical side, http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1992PhDT.......130L gives in an abstract of a 1982 thesis of Anzhi Lai an experimental upper bound on the electron charge radius of ~ 10^{-16} cm (But I haven't seen the thesis.)

According to (7.12) in Phys. Rev. D 62, 113012 (2000), the charge radius of neutrinos, another pointlike particle, computed from the standard model to 1 loop order, is in the range of 4...6 10^-14 cm for the three neutrino species.

hep-ph/0109138 reviews the experimental limits on the sizes of fundamental particles.

Everything discussed above assumes that the particle is isolated and at rest. The form factor tells about its response to external electromagnetic fields and about the electromagnetic field it is accompanied by.

But the form factor contains nothing at all about interaction- or state-dependent information. The interaction-dependent information is instead coded in an external potential or a multiparticle formulation, and the state-dependent information is coded in the wave function or density matrix, which (at any given time) is independent of the Hamiltonian.

Also, the information contained in the form factor is only about the free particle in the rest system, defined by a state in which momentum and orbital angular momentum vanish identically. In an external potential, or in a state where momentum (or orbital angular momentum) doesn't vanish, the charge density (and the resulting charge radius) can differ arbitrarily much from the charge density (and charge radius) at rest. For example, for a hydrogen electron in the ground state, the charge density (which must essentially cancel the proton's Coulomb field far away from the atom) is significant in a region of diameter about 10^{-11} cm (a small multiple of the Bohr radius), while the charge radius at rest is probably (in view of the above partial results) < 10^{-13} cm.

In all cases, the charge distribution is defined as the expectation of the charge density operator of the corresponding quantum field. For molecules, this charge distribution is the computational target of much of quantum chemistry, and defines the shape of a molecule. The shape of a particle determined by the form factor therefore corresponds to the equilibrium shape a molecule takes in its rest frame in the absence of forces, i.e., in its ground state, while the state-dependent shape corresponds to the much less predictable shape of a molecule interacting with its environment.


CODATA Recommended Values of the Fundamental Physical Constants: 2010 lists as ''classical electron radius'' the value 2.8179403267(27) fm. (1fm=10^{-15}m)

For nuclear radii see, e.g., p.17 of CODATA Recommended Values of the Fundamental Physical Constants: 2010

For (constituent) quark sizes see, e.g., NN Scattering and Nucleon Quark Core


Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ