In theoretical physics, a mixed anomaly is an example of an

anomaly Anomaly may refer to: Science Natural *Anomaly (natural sciences) ** Atmospheric anomaly ** Geophysical anomaly Medical * Congenital anomaly (birth defect), a disorder present at birth ** Physical anomaly, a deformation of an anatomical struct ...

: it is an effect of quantum mechanics — usually a one-loop diagram — that implies that the classically valid general covariance and

gauge symmetry 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 groups) ...

of a theory of general relativity combined with gauge fields and

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 ...

ic fields cannot be preserved simultaneously in the quantum theory. The adjective "mixed" usually refers to a mixture of a gravitational anomaly and gauge anomaly, but may also refer to a mixture of two different gauge groups tensored together, like the ''SU(2)'' and the ''U(1)'' of the

Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...

. The anomaly usually appears as a Feynman diagram with a chiral

running in the loop (a polygon) with ''n−k'' external gravitons and ''k'' external

gauge boson In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge ...

s attached to the loop where

n=1+D/2

where

D

is the spacetime dimension. Chiral fermions only occur in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams. General covariance and gauge symmetries are very important symmetries for the consistency of the whole theory, and therefore all gravitational, gauge, and mixed anomalies must cancel out.

References

See also