Boundary anomalous dimensions from BCFT: O(N) -symmetric ϕ2n theories with a boundary and …

We investigate the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>ϕ</a:mi><a:mrow><a:mn>2</a:mn><a:mi>n</a:mi></a:mrow></a:msup></a:math> deformations of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi mathvariant="normal">O</c:mi><c:mo stretchy="false">(</c:mo><c:mi>N</c:mi><c:mo stretchy="false">)</c:mo></c:mrow></c:math>-symmetric (generalized) free theories with a flat boundary, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>n</h:mi><h:mo>≥</h:mo><h:mn>2</h:mn></h:math> is an integer. The generalized free theories refer to the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mo>□</j:mo><j:mi>k</j:mi></j:msup></j:math> free scalar theories with a higher derivative kinetic term, which is related to the multicritical generalizations of the Lifshitz type. We assume that the (generalized) free theories and the deformed theories have boundary conformal symmetry and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mi mathvariant="normal">O</l:mi><l:mo stretchy="false">(</l:mo><l:mi>N</l:mi><l:mo stretchy="false">)</l:mo></l:mrow></l:math> global symmetry. The leading anomalous dimensions of some boundary operators are derived from the bulk multiplet recombination and analyticity constraints. We find that the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>ε</q:mi><q:mrow><q:mn>1</q:mn><q:mo>/</q:mo><q:mn>2</q:mn></q:mrow></q:msup></q:math> expansion in the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mi>ϕ</s:mi><s:mn>6</s:mn></s:msup></s:math>-tricritical version of the special transition extends to other multicritical cases with larger odd integer <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>n</u:mi></u:math>, and most of the higher derivative cases involve a noninteger power expansion in <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>ε</w:mi></w:math>. Using the analytic bootstrap, we further verify that the multiplet-recombination results are consistent with boundary crossing symmetry.