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Anomaly (physics)
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==Gauge anomalies== {{Main article|Gauge anomaly}} Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel unphysical degrees of freedom with a negative norm (such as a [[photon]] polarized in the time direction). An attempt to cancel them—i.e., to build theories [[consistent]] with the gauge symmetries—often leads to extra constraints on the theories (such is the case of the [[gauge anomaly]] in the [[Standard Model]] of particle physics). Anomalies in [[gauge theory|gauge theories]] have important connections to the [[topology]] and [[geometry]] of the [[gauge group]]. Anomalies in gauge symmetries can be calculated exactly at the one-loop level. At tree level (zero loops), one reproduces the classical theory. [[Feynman diagrams]] with more than one loop always contain internal [[boson]] propagators. As bosons may always be given a mass without breaking gauge invariance, a [[Pauli–Villars regularization]] of such diagrams is possible while preserving the symmetry. Whenever the regularization of a diagram is consistent with a given symmetry, that diagram does not generate an anomaly with respect to the symmetry. Vector gauge anomalies are always [[chiral anomaly|chiral anomalies]]. Another type of gauge anomaly is the [[gravitational anomaly]].
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