Flavor structure of the energy-momentum tensor form factors of the proton

The energy-momentum tensor form factors furnish information on the mechanics of the proton. It is essential to compute the generalized isovector-vector form factors to examine the flavor structure of the energy-momentum tensor form factors. The flavor-decomposed form factors reveal the internal structure of the proton. The up quark dominates over the down quark for the mass and spin of the proton, whereas the down quark takes over the up quark for the $D$-term form factor. We investigate for the first time the isovector $\bar{c}(t)$ form factor of the proton and its physical implications. The flavor-decomposed $\bar{c}(t)$ form factors of the proton unveil how the up-quark contribution is exactly canceled by the down-quark contribution inside a proton within the framework of the pion mean-field approach. While the proton $\bar{c}(t)$ form factor does not contribute to the proton mass, its flavor structure sheds light on how the strong force fields due to the $\bar{c}(t)$ form factor characterize the stability of the proton.