How to improve the maps of magnetic helicity injection in active regions?

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Magnetic helicity, a topological quantity which measures the twist, the writhe and the shear of a magnetic field, has recently appeared as a key quantity to understand some mechanisms of the solar activity such as Coronal Mass Ejections and flare onset. It is thus becoming of major importance to be able to compute magnetic helicity in active regions. Computing photospheric maps of the injection of magnetic helicity provides new spatial information that helps us to understand basic properties of solar activity, such as where and how magnetic helicity is injected. Several helicity flux density maps have been published for different active regions. Unfortunately, the classical helicity flux density is not a correct physical quantity and it does induce spurious signals (fake polarities) which mask the real injection of helicity. To map the real helicity injection, the knowledge of the complete connectivity of the field lines is fundamental. Even without the connectivity, improved helicity flux density maps can be derived. They have fake polarities which are lower by more than a factor 10 than the previous incorrect maps. Rather than a mixture of negative and positive injection patterns, they show almost unipolar injection on the active region scale. This leads to a completely new way of understanding the dynamics of active regions, in the frame of magnetic helicity studies.