Anomaly matching condition
Encyclopedia
In quantum field theory
, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly
by using the degrees of freedom of the theory at some energy scale, must not depend on what scale is chosen for the calculation. It is also known as 't Hooft UV-IR anomaly matching condition, where UV stand for the high-energy limit, and IR for the low-energy limit of the theory.
For example, one may calculate the anomaly of a global U(1) current by considering which fermion
ic degrees of freedom
the theory has. One may use either the degrees of freedom
at the far IR (i.e. low energy limit) or the degrees of freedom at the far UV (i.e. high energy limit) in order to calculate the anomaly. In the former case one should only consider mass
less fermion
s which may be composite particles, while in the latter case one should only consider the elementary fermion
s of the underlying short-distance theory. In both cases, the answer must be the same.
One proves this condition by the following procedure: we may add to the theory a gauge field which couples
to the current related with this symmetry, as well as chiral fermion
s which couple
only to this gauge field, and cancel the anomaly (so that the gauge symmetry will remain non-anomalous
, as needed for consistency).
In the limit were the coupling constant
s we have added go to zero, one gets back to the original theory, plus the fermions we have added; the latter remain good degrees of freedom at every energy scale, as they are free fermions at this limit. The gauge symmetry anomaly can be computed at any energy scale, and must always be zero, so that the theory is consistent. One may now get the anomaly of the symmetry in the original theory by subtracting the free fermions we have added, and the result is independent of the energy scale.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
by using the degrees of freedom of the theory at some energy scale, must not depend on what scale is chosen for the calculation. It is also known as 't Hooft UV-IR anomaly matching condition, where UV stand for the high-energy limit, and IR for the low-energy limit of the theory.
For example, one may calculate the anomaly of a global U(1) current by considering which fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
ic degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
the theory has. One may use either the degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
at the far IR (i.e. low energy limit) or the degrees of freedom at the far UV (i.e. high energy limit) in order to calculate the anomaly. In the former case one should only consider mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
less fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s which may be composite particles, while in the latter case one should only consider the elementary fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s of the underlying short-distance theory. In both cases, the answer must be the same.
One proves this condition by the following procedure: we may add to the theory a gauge field which couples
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
to the current related with this symmetry, as well as chiral fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s which couple
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
only to this gauge field, and cancel the anomaly (so that the gauge symmetry will remain non-anomalous
Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
, as needed for consistency).
In the limit were the coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
s we have added go to zero, one gets back to the original theory, plus the fermions we have added; the latter remain good degrees of freedom at every energy scale, as they are free fermions at this limit. The gauge symmetry anomaly can be computed at any energy scale, and must always be zero, so that the theory is consistent. One may now get the anomaly of the symmetry in the original theory by subtracting the free fermions we have added, and the result is independent of the energy scale.