Vacuum transitions with the Gauss-Bonnet term in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi></mml:math> dimensions

In our previous paper [; ], we proposed a probabilistic argument to explain the reason why the cosmological constant is very small in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>4</a:mn><a:mi>D</a:mi></a:math>. We can ask a question: if the behavior of tunneling exponent <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>B</c:mi></c:math> can be generalized to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>D</e:mi></e:math>-dimension. Moreover, in higher dimensional theory motivated by string theory the Gauss-Bonnet term plays an important role. Therefore, in this paper, we generalize our result in previous paper [; ] to arbitrary <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>D</g:mi></g:math> dimensions including the Gauss-Bonnet term. As a result, we have two main results. We find that the Euclidean action of the bounce, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>B</i:mi></i:math>, describing the decay of a de Sitter vacuum, is proportional to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msubsup><k:mi>k</k:mi><k:mo>+</k:mo><k:mrow><k:mo>−</k:mo><k:mo stretchy="false">(</k:mo><k:mi>D</k:mi><k:mo>−</k:mo><k:mn>2</k:mn><k:mo stretchy="false">)</k:mo></k:mrow></k:msubsup></k:math>, which has a pole as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msubsup><o:mi>k</o:mi><o:mo>+</o:mo><o:mn>2</o:mn></o:msubsup><o:mo stretchy="false">→</o:mo><o:mn>0</o:mn></o:math> where <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msubsup><r:mi>k</r:mi><r:mo>+</r:mo><r:mn>2</r:mn></r:msubsup></r:math> is the curvature of the parent vacuum. This result is similar to the result in <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mn>4</t:mn><t:mi>D</t:mi></t:math>. The other result is that we find a new decay channel, describing up-tunneling from anti–de Sitter into de Sitter. The meaning of this new decay channel in the string landscape should be explored in the future.