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The isosinglet combination of the chiral-odd twist-3 distribution function$e^u(x)+e^d(x)$ of the nucleon has outstanding properties that its first momentis proportional to the well-known $\pi N$ sigma-term and that it contains a$\delta$-function singularity at $x=0$. These two features are inseparablyconnected in that the above sum rule would be violated, if there is no such asingularity in $e^u(x)+e^d(x)$. Very recently, we found that the physicalorigin of this $\delta$-function singularity can be traced back to thelong-range quark-quark correlation of scalar type, which signals thespontaneous chiral symmetry breaking of the QCD vacuum. The main purpose of thepresent paper is to give complete theoretical predictions for the chiral-oddtwist-3 distribution function $e^a(x)$ of each flavor $a$ on the basis of thechiral quark soliton model, without recourse to the derivative expansion typeapproximation. These theoretical predictions are then compared with theempirical information extracted from the CLAS data of the semi-inclusive DISprocesses by assuming the Collins mechanism only. A good agreement with theCLAS data is indicative of a sizable violation of the $\pi N$ sigma-term sumrule, or equivalently, the existence of a $\delta$-function singularity in$e^u(x) + e^d(x)$.Comment: 35 pages, 10 figure

πN sigma term and chiral-odd twist-3 distribution function e(x) of the nucleon in the chiral quark soliton model