All actions for free massive higher-spin fields

Massive higher-spin states/fields appear in the effective description of various systems from hadrons and nuclei to black holes, whenever the point-particle approximation is justified, as well as in the bottom-up approaches to the quantum gravity problem. In four dimensions the actions for massive higher-spin fields utilize either the Singh-Hagen/Zinoviev set of auxiliary fields or a single chiral field, which is an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>s</a:mi><a:mi>l</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mo>,</a:mo><a:mi mathvariant="double-struck">C</a:mi><a:mo stretchy="false">)</a:mo></a:math> spin-tensor of type <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mo stretchy="false">(</f:mo><f:mn>2</f:mn><f:mi>s</f:mi><f:mo>,</f:mo><f:mn>0</f:mn><f:mo stretchy="false">)</f:mo></f:math>, generalizing the Chalmers-Siegel approach. We show that these two actions are on the opposite ends of a discrete family of actions where the physical field is a spin-tensor of type <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mo stretchy="false">(</j:mo><j:mi>s</j:mi><j:mo>+</j:mo><j:mi>k</j:mi><j:mo>,</j:mo><j:mi>s</j:mi><j:mo>−</j:mo><j:mi>k</j:mi><j:mo stretchy="false">)</j:mo></j:math>. The <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mo stretchy="false">(</n:mo><n:mn>2</n:mn><n:mi>s</n:mi><n:mo>−</n:mo><n:mn>1</n:mn><n:mo>,</n:mo><n:mn>1</n:mn><n:mo stretchy="false">)</n:mo></n:math>- and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mo stretchy="false">(</r:mo><r:mn>2</r:mn><r:mi>s</r:mi><r:mo>−</r:mo><r:mn>2</r:mn><r:mo>,</r:mo><r:mn>2</r:mn><r:mo stretchy="false">)</r:mo></r:math>-cases generalize the Proca and the Fierz-Pauli actions, respectively, to all spins. A similar family of second-order actions exists for fermionic higher-spin fields.