Proposal to look for the anomalous isotopic symmetry breaking in central diffractive production of the $f_1(1285)$ and $a^0_0(980)$ resonances at the LHC
At very high energies, and in the central region ($x_F\simeq0$), the double-Pomeron exchange mechanism gives the dominant contribution to the production of hadrons with the positive $C$ parity and isospin $I=0$. Therefore, the observation of resonances in the states with $I=1$ will be indicative of their production or decay with the isotopic symmetry breaking. Here, we bear in mind the cases of the anomalous breaking of the isotopic symmetry, i.e., when the cross section of the process breaking the isospin is not of the order of $10^{-4}$ of the cross section of the allowed process but of the order of $1\%$. The paper draws attention to the reactions $pp\to p(f_1(1285)/f_1(1420))p\to p(π^+π^-π^0)p$ and $pp\to p(K\bar K)p\to p(a^0_0(980))p\to p(ηπ^0)p$ in which a similar situation can be realized, owing to the $K\bar K$ loop mechanisms of the breaking of isotopic symmetry. We note that there is no visible background in the $π^+π^-π^0$ and $ηπ^0$ channels. Observation of the process $pp\to p(f_1(1285))p\to p(π^+π^-π^0)p$ would be a confirmation of the first results from the VES and BESIII detectors, indicating the very large isospin breaking in the decay $f_1(1285))\toπ^+π^-π^0$.