A search for a Higgs boson decaying into a pair of electrons or muons and a photon is described. Higgs boson decays to a Z boson and a photon (H $\to$ Z$\gamma\to\ell\ell\gamma$, $\ell =$ e or $\mu$), or to two photons, one of which has an internal conversion into a muon pair (H $\to\gamma^{*}\gamma\to\mu\mu\gamma$) were considered. The analysis is performed using a data set recorded by the CMS experiment at the LHC from proton-proton collisions at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. No significant excess above the background prediction has been found. Limits are set on the cross section for a standard model Higgs boson decaying to opposite-sign electron or muon pairs and a photon. The observed limits on cross section times the corresponding branching fractions vary between 1.4 and 4.0 (6.1 and 11.4) times the standard model cross section for H $\to\gamma^{*}\gamma\to\mu\mu\gamma$ (H $\to$ Z$\gamma\to\ell\ell\gamma$) in the 120-130 GeV mass range of the $\ell\ell\gamma$ system. The H $\to\gamma^*\gamma\to\mu\mu\gamma$ and H $\to$ Z$\gamma\to\ell\ell\gamma$ analyses are combined for $m_\mathrm{H} =$ 125 GeV, obtaining an observed (expected) 95% confidence level upper limit of 3.9 (2.0) times the standard model cross section.
Differential Higgs boson (H) production cross sections are sensitive probes for physics beyond the standard model. New physics may contribute in the gluon-gluon fusion loop, the dominant Higgs boson production mechanism at the LHC, and manifest itself through deviations from the distributions predicted by the standard model. Combined spectra for the H $\to$ $\gamma\gamma$, H $\to$ ZZ, and H $\to$ $\mathrm{b\overline{b}}$ decay channels and the inclusive Higgs boson production cross section are presented, based on proton-proton collision data recorded with the CMS detector at $\sqrt{s} =$ 13 TeV corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The transverse momentum spectrum is used to place limits on the Higgs boson couplings to the top, bottom, and charm quarks, as well as its direct coupling to the gluon field. No significant deviations from the standard model are observed in any differential distribution. The measured total cross section is 61.1 $\pm$ 6.0 (stat) $\pm$ 3.7 (syst) pb, and the precision of the measurement of the differential cross section of the Higgs boson transverse momentum is improved by about 15% with respect to the H $\to$ $\gamma\gamma$ channel alone.
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