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Semi-infinite $d$-dimensional systems with an $m$-axial bulk Lifshitz pointare considered whose ($d-1$)-dimensional surface hyper-plane is orientedperpendicular to one of the $m$ modulation axes. An $n$-component $\phi^4$field theory describing the bulk and boundary critical behaviour when (i) theHamiltonian can be taken to have O(n) symmetry and (ii) spatial anisotropiesbreaking its Euclidean symmetry in the $m$-dimensional coordinate subspace ofpotential modulation directions may be ignored is investigated. Thelong-distance behaviour at the ordinary surface transition is mapped onto afield theory with the boundary conditions that both the order parameter$\bm{\phi}$ and its normal derivative $\partial_n\bm{\phi}$ vanish at thesurface plane. The boundary-operator expansion is utilized to study theshort-distance behaviour of $\bm{\phi}$ near the surface. Its leadingcontribution is found to be controlled by the boundary operator$\partial_n^2\bm{\phi}$. The field theory is renormalized for dimensions $d$below the upper critical dimension $d^*(m)=4+m/2$, with a corresponding surfacesource term $\propto \partial_n^2\bm{\phi}$ added. The anomalous dimension ofthis boundary operator is computed to first order in $\epsilon=d^*-d$. Theresult is used in conjunction with scaling laws to estimate the value of thesingle independent surface critical exponent$\beta_{\mathrm{L}1}^{(\mathrm{ord},\perp)}$ for $d=3$. Our estimate for thecase $m=n=1$ of a uniaxial Lifshitz point in Ising systems is in reasonableagreement with published Monte Carlo results.

Boundary critical behaviour atm-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis