Searching for Odderon in Central Exclusive Processes at the LHC

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There seem to be recently an evidence for $C = -1$ odderon exchange in proton-proton elastic scattering at high energies. Here we discuss three different central-exclusive-production processes $pp \to pp (\phi \to K^{+}K^{-})$, $pp \to pp (\phi \to \mu^{+}\mu^{-})$, and $pp \to pp \phi \phi \to pp K^{+}K^{-}K^{+}K^{-}$, as a possible source of information for soft odderon exchange. The theoretical results are calculated within the tensor-pomeron and vector-odderon model for soft reactions. We include absorptive corrections at the amplitude level. To describe the low-energy data measured in the past by the WA102 collaboration we consider also subleading processes with reggeized vector-meson exchanges and reggeons. We discuss possible evidences for odderon exchange at the low energies and try to make predictions at the LHC.