This is the last in a series of papers on the topological susceptibility in the interacting instanton liquid model ( IILM ) .
We will derive improved finite temperature interactions to study the thermodynamic limit of grand canonical Monte Carlo simulations in the quenched and unquenched case with light , physical quark masses .
In particular , we will be interested in chiral symmetry breaking .
The paper culminates by giving , for the first time , a well-motivated temperature-dependent axion mass .
Especially , this work finally provides a computation of the axion mass in the low temperature regime , m ^ { 2 } _ { a } f ^ { 2 } _ { a } = 1.46 10 ^ { -3 } \Lambda ^ { 4 } \frac { 1 + 0.50 T / \Lambda } { 1 + \left ( 3. % 53 T / \Lambda \right ) ^ { 7.48 } } .
It connects smoothly to the high temperature dilute gas approximation ; the latter is improved by including quark threshold effects .
To compare with earlier studies , we also provide the usual power-law m ^ { 2 } _ { a } = \frac { \alpha _ { a } \Lambda ^ { 4 } } { f _ { a } ^ { 2 } ( T / \Lambda ) ^ { n } } , where \Lambda = 400 \mathrm { MeV } , n = 6.68 and \alpha = 1.68 10 ^ { -7 } .