We suggest that the exclusive Higgs + light (or b)-jet production at the LHC, $pp to h+j(j_b)$, is a rather sensitive probe of the light-quarks Yukawa couplings and of other forms of new physics (NP) in the Higgs-gluon $hgg$ and quark-gluon $qqg$ interactions. We study the Higgs $p_T$-distribution in $pp to h+j(j_b) to gamma gamma + j(j_b)$, i.e., in $h+j(j_b)$ production followed by the Higgs decay $h to gamma gamma$, employing the ($p_T$-dependent) signal strength formalism to probe various types of NP which are relevant to these processes and which we parameterize either as scaled Standard Model (SM) couplings (the kappa-framework) and/or through new higher dimensional effective operators (the SMEFT framework). We find that the exclusive $h+j(j_b)$ production at the 13 TeV LHC is sensitive to various NP scenarios, with typical scales ranging from a few TeV to ${cal O}(10)$ TeV, depending on the flavor, chirality and Lorentz structure of the underlying physics.
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