In
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, the vertex function describes the coupling between a
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
and an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
beyond the leading order of
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. In particular, it is the
one particle irreducible correlation function involving the
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
, the antifermion
, and the
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
A.
Definition
The vertex function
can be defined in terms of a
functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
of the
effective action
In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective act ...
S
eff as
:
The dominant (and classical) contribution to
is the
gamma matrix
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
, which explains the choice of the letter. The vertex function is constrained by the symmetries of quantum electrodynamics —
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ve ...
;
gauge invariance
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
or the
transversality of the photon, as expressed by the
Ward identity
Ward may refer to:
Division or unit
* Hospital ward, a hospital division, floor, or room set aside for a particular class or group of patients, for example the psychiatric ward
* Prison ward, a division of a penal institution such as a pris ...
; and invariance under
parity — to take the following form:
:
where
,
is the incoming four-momentum of the external photon (on the right-hand side of the figure), and F
1(q
2) and F
2(q
2) are ''
form factors'' that depend only on the momentum transfer q
2. At tree level (or leading order), F
1(q
2) = 1 and F
2(q
2) = 0. Beyond leading order, the corrections to F
1(0) are exactly canceled by the
field strength renormalization
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. The form factor F
2(0) corresponds to the
anomalous magnetic moment
In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
''a'' of the fermion, defined in terms of the
Landé g-factor
In physics, the Landé ''g''-factor is a particular example of a ''g''-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921.
In atomic physics, the Landé ''g ...
as:
:
References
*
*
*
External links
*
Quantum electrodynamics
Quantum field theory
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