Context : The origin of radio halos in galaxy clusters is still unknown and is the subject of a vibrant debate both from the observational and theoretical point of view .
In particular the amount and the nature of non-thermal plasma and of the magnetic field energy density in clusters hosting radio halos is still unclear .

Aims : The aim of this paper is to derive an estimate of the pressure ratio X = P _ { non - th } / P _ { th } between the non-thermal and thermal plasma in radio halo clusters that have combined radio , X-ray and SZ effect observations .

Methods : From the simultaneous P _ { 1.4 } - L _ { X } and P _ { 1 , 4 } - Y _ { SZ } correlations for a sample of clusters observed with Planck , we derive a correlation between Y _ { SZ } and L _ { X } that we use to derive a value for X .
This is possible since the Compton parameter Y _ { SZ } is proportional to the total plasma pressure in the cluster ( that we characterize as the sum of the thermal and non-thermal pressure ) while the X-ray luminosity L _ { X } is proportional only to the thermal pressure of the intracluster plasma .

Results : Our results indicate that the average ( best fit ) value of the pressure ratio in a self-similar cluster formation model is X = 0.55 \pm 0.05 in the case of an isothermal \beta -model with \beta = 2 / 3 and a core radius r _ { c } = 0.3 \cdot R _ { 500 } holding on average for the cluster sample .
We also show that the theoretical prediction for the Y _ { SZ } - L _ { X } correlation in this model has a slope that is steeper than the best fit value for the available data .
The agreement with the data can be recovered if the pressure ratio X decreases with increasing X-ray luminosity as L _ { X } ^ { -0.96 } .

Conclusions : We conclude that the available data on radio halo clusters indicate a substantial amount of non-thermal pressure in cluster atmospheres whose value must decrease with increasing X-ray luminosity , or increasing cluster mass ( temperature ) .
This is in agreement with the idea that non-thermal pressure is related to non-thermal sources of cosmic rays that live in cluster cores and inject non-thermal plasma in the cluster atmospheres that is subsequently diluted by the ICM acquired during cluster collapse , and has relevant impact for further studies of high-energy phenomena in galaxy clusters .