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We explore the possibility of observing odderon exchange in the $pp \to ppK^{+}K^{-}$ and $pp \to pp \mu^{+}\mu^{-}$ reactions at the LHC. We considerthe central exclusive production (CEP) of the $\phi(1020)$ resonance decayinginto $K^{+} K^{-}$ and $\mu^{+}\mu^{-}$. We compare the purely diffractivecontribution (odderon-pomeron fusion) to the photoproduction contribution(photon-pomeron fusion). The theoretical results are calculated within thetensor-pomeron and vector-odderon model for soft reactions. We includeabsorptive corrections at the amplitude level. In order to fix the couplingconstants for the photon-pomeron fusion contribution we discuss the reactions$\gamma p \to \omega p$ and $\gamma p \to \phi p$ including $\phi$-$\omega$mixing. We compare our results for these reactions with the available data,especially those from HERA. Our coupling constants for thepomeron-odderon-$\phi$ vertex are taken from an analysis of the WA102 data forthe $p p \to p p \phi$ reaction. We show that the odderon-exchange contributionsignificantly improves the description of the $pp$ azimuthal correlations andthe $dP_{t}$ "glueball-filter variable" dependence of $\phi$ CEP measured byWA102. To describe the low-energy data more accurately we consider alsosubleading processes with reggeized vector-meson exchanges. However, they donot play a significant role at the LHC. We present predictions for two possibletypes of measurements: at midrapidity and with forward measurement of protons(relevant for ATLAS-ALFA or CMS-TOTEM), and at forward rapidities and withoutmeasurement of protons (relevant for LHCb). We discuss the influence ofexperimental cuts on the integrated cross sections and on various differentialdistributions. With the corresponding LHC data one should be able to get adecisive answer concerning the presence of an odderon-pomeron fusioncontribution in single $\phi$ CEP.

Searching for the odderon in $pp \to pp K^{+}K^{-}$ and $pp \to pp \mu^{+}\mu^{-}$ reactions in the $\phi(1020)$ resonance region at the LHC