Boundary anomalous dimensions from BCFT: $φ^{3}$ theories with a boundary and higher-derivative generalizations

Abstract: We consider the bulk $φ^3$ deformation of the free boundary conformal field theory in the $ε$ expansion. We determine the leading corrections to the scaling dimensions of boundary fundamental operators and some boundary operator expansion coefficients. Our procedure combines the conformal multiplet recombination with the boundary crossing symmetry. The results cover both the single field case and the multi-field case with $S_{N+1}$ global symmetry, which are associated with the Yang-Lee model and the $(N+1)$-state Potts model respectively. These semi-infinite models describe branched polymers, percolation, and spanning forest at a surface. We generalize these results to some higher derivative theories. In addition, we study the $φ^{2n+1}$ theories with $n>1$, but only obtain some boundary operator expansion coefficients.

• The paper's main contribution is determining boundary anomalous dimensions in φ³ BCFTs through multiplet recombination and crossing symmetry.
• It computes ε-expansion corrections for both single-field Yang-Lee and multi-field Potts models, linking scaling dimensions to critical phenomena.
• The study extends results to higher-derivative and multicritical cases, opening avenues for refined bootstrap and RG analyses.

Boundary Anomalous Dimensions in $\phi^3$ BCFTs and Higher-Derivative Extensions

Overview and Motivation

The paper investigates the determination of boundary anomalous dimensions and operator expansion coefficients in boundary conformal field theories (BCFTs) deformed by a $\phi^3$ interaction, focusing on both single- and multi-field cases and their higher-derivative generalizations. The analysis leverages the interplay between conformal multiplet recombination and boundary crossing symmetry. The physical relevance spans critical phenomena such as Yang-Lee edge singularity, branched polymers, percolation, and the Potts model, especially in the presence of boundaries. These results are further generalized to higher-derivative kinetic terms (e.g., Lifshitz-type theories) and multicritical odd interactions $\phi^{2n+1}$, providing a comprehensive axiomatic framework for accessing BCFT data in these settings.

Technical Approach

Multiplet Recombination and Boundary Crossing Symmetry

The key method involves using multiplet recombination to constrain the structure of BCFT operator content in the presence of the $\phi^3$ interaction. Specifically, the free multiplets recombine in the interacting theory, allowing identification of new boundary contributions and their scaling dimensions. The multiplet recombination approach is distinguished from bootstrap methods, offering an axiomatic route to deduce conformal data in the $\epsilon$-expansion.

Boundary crossing symmetry is implemented by analyzing the two-point function $(\phi(x)\phi(y))$ on the upper half-space, which admits dual bulk and boundary channel expansions in terms of conformal blocks. The crossing equation—matching these expansions—yields strong constraints on operator dimensions and boundary operator expansion coefficients (BOEs). The combination of both techniques is essential: multiplet recombination fixes the bulk-to-boundary correspondence, while the crossing equation determines the residual unknowns.

Cases Studied

Yang-Lee/Branched Polymer Model (Single-Field)

For the single-field $\phi^3$ theory, corresponding to the Yang-Lee model and branched polymers, leading $\epsilon$-expansion corrections to scaling dimensions and BOEs are computed systematically, covering both ordinary (Neumann) and special (Dirichlet) boundary conditions.

Potts Model (Multi-Field, $S_{N+1}$ Symmetry)

Extending to the $S_{N+1}$ symmetric multi-field case, connections to the $\phi^3$0-state Potts model, percolation ($\phi^3$1), and spanning forest ($\phi^3$2) are detailed. Here, the tensor structure and representation theory influence boundary operator content and the crossing analysis. Notably, the large $\phi^3$3 limit behavior in the presence of boundaries does not simply decouple into single-field models, reflecting the nonlocal nature of the boundary as a probe.

Higher-Derivative and Multicritical Generalizations

Explicit calculations are performed for $\phi^3$4 kinetic terms ($\phi^3$5), extending results to Lifshitz-type fixed points. The analytic structure and boundary scaling dimensions for these generalizations are obtained for the first time, revealing new IR conformal fixed points and their boundary data. Partial results are also derived for multicritical odd interactions $\phi^3$6 ($\phi^3$7), although boundary anomalous dimensions remain undetermined due to insufficient crossing constraints without further input.

Numerical Results and Claims

Strong numerical results are derived for boundary anomalous dimensions and BOE coefficients in the $\phi^3$8-expansion, including:

• Yang-Lee model (single-field): The anomalous dimension shifts for boundary fundamental operators are obtained for both Neumann and Dirichlet cases, with explicit dependence on $\phi^3$9 (e.g., $\phi^{2n+1}$0 for Dirichlet).
• Potts model (multi-field): The scaling dimension corrections in terms of $\phi^{2n+1}$1, elucidating percolation ($\phi^{2n+1}$2) and spanning forest ($\phi^{2n+1}$3), demonstrate a precise quantitative link with surface critical exponents.
• Higher-derivative cases: First-order corrections to boundary anomalous dimensions are computed for $\phi^{2n+1}$4, generalizing canonical field-theory results to systems near Lifshitz points.
• BOE and OPE coefficients: Computed ratios and coefficients (e.g., $\phi^{2n+1}$5 and $\phi^{2n+1}$6) are in accord with prior bulk determinations and new boundary computations, demonstrating the effectiveness and consistency of the methodology.

The paper asserts that the large $\phi^{2n+1}$7 limit of the Potts BCFT with a boundary does not reproduce the single-field result, contrasting with bulk behavior. This has significant implications for boundary universality and the role of nonlocal probes.

Implications and Future Directions

Practical Impact

The results are directly applicable to the understanding of surface critical behavior in statistical mechanics models (e.g., branched polymers, percolation, Potts model), providing analytic control in perturbatively non-unitary settings. The higher-derivative generalizations broaden the landscape of boundary critical phenomena, particularly for systems relevant to Lifshitz points and multicriticality.

Theoretical Advances

Methodologically, the combination of multiplet recombination and boundary crossing symmetry refines axiomatic approaches to BCFT, especially in non-unitary and higher-derivative cases where perturbation theory or positivity-based numerical bootstrap is insufficient. The explicit computation of boundary data at new fixed points establishes a foundation for further exploration of boundary and interface CFTs.

The inability to fix boundary anomalous dimensions in $\phi^{2n+1}$8 theories (for $\phi^{2n+1}$9) signals the need to integrate additional Lagrangian or RG input or to develop new bootstrap techniques not reliant on sign-definite OPE coefficients.

Future Directions

Speculative avenues include:

• Incorporation of Lagrangian information in the bootstrap, possibly refining conformal predictions in non-unitary and multicritical settings.
• Extension to interface CFTs (using folding tricks), which may elucidate RG domain wall properties and relate UV/IR fixed points.
• Development of new bootstrap techniques and analytic methods for boundary CFTs without positivity constraints, as well as adaptation to defect and interface scenarios.
• Detailed exploration of operator content and OPE structures for multicritical and higher-derivative models, potentially via bulk-bulk-boundary three-point functions.

Conclusion

This paper provides a rigorous treatment of boundary anomalous dimensions and operator expansion coefficients in $\phi^3$0 BCFTs and their higher-derivative extensions. Results cover both single-field (Yang-Lee/branched polymer) and multi-field (Potts/percolation/spanning forest) cases, analytically accessing boundary critical data in the $\phi^3$1-expansion. The methodology is robust, combining multiplet recombination and boundary crossing, and the implications range from practical modeling of surface critical phenomena to foundational advances in CFT bootstrap techniques. Future work is anticipated to further generalize these results and address unresolved questions for multicritical odd interaction theories and interface CFTs.