Bosonic and fermionic Love numbers of a static acoustic black hole

We compute static ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>ω</a:mi> <a:mo>→</a:mo> <a:mn>0</a:mn> </a:mrow> </a:math> ) tidal Love numbers for scalar ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>s</b:mi> <b:mo>=</b:mo> <b:mn>0</b:mn> </b:mrow> </b:math> ) and Dirac ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>s</c:mi> <c:mo>=</c:mo> <c:mn>1</c:mn> <c:mo>/</c:mo> <c:mn>2</c:mn> </c:mrow> </c:math> ) perturbations of static acoustic black holes (ABHs) in <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mo>(</d:mo> <d:mn>3</d:mn> <d:mo>+</d:mo> <d:mn>1</d:mn> <d:mo>)</d:mo> </d:mrow> </d:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mo>(</e:mo> <e:mn>2</e:mn> <e:mo>+</e:mo> <e:mn>1</e:mn> <e:mo>)</e:mo> </e:mrow> </e:math> dimensions, respectively. By imposing horizon regularity condition and matching to the large-radius expansion, we extract the ratio between decaying and growing modes. It turns out that in <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mo>(</f:mo> <f:mn>3</f:mn> <f:mo>+</f:mo> <f:mn>1</f:mn> <f:mo>)</f:mo> </f:mrow> </f:math> dimensions the scalar Love number is generically nonzero for ABHs, while the Fermionic Love numbers follow a universal power-law form <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:msubsup> <g:mi>F</g:mi> <g:mrow> <g:mi>ℓ</g:mi> <g:mi>m</g:mi> </g:mrow> <g:mrow> <g:mo>±</g:mo> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mn>2</g:mn> </g:mrow> </g:msubsup> <g:mo>=</g:mo> <g:mo>±</g:mo> <g:msup> <g:mn>4</g:mn> <g:mrow> <g:mo>−</g:mo> <g:mo>(</g:mo> <g:mi>ℓ</g:mi> <g:mo>+</g:mo> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mn>2</g:mn> <g:mo>)</g:mo> </g:mrow> </g:msup> </g:mrow> </g:math> . In <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mo>(</h:mo> <h:mn>2</h:mn> <h:mo>+</h:mo> <h:mn>1</h:mn> <h:mo>)</h:mo> </h:mrow> </h:math> dimensions, the scalar field exhibits a strange logarithmic structure, causing the Bosonic Love number to vanish for even <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>m</i:mi> </i:math> but remain nontrivial for odd <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>m</j:mi> </j:math> . In contrast, the Fermionic Love number in this case retains a simple power-law form <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:msub> <k:mi>F</k:mi> <k:mi>m</k:mi> </k:msub> <k:mo>=</k:mo> <k:msup> <k:mn>4</k:mn> <k:mrow> <k:mo>−</k:mo> <k:mi>m</k:mi> </k:mrow> </k:msup> </k:mrow> </k:math> and is generically nonzero. These results provide insights into tidal response in analog-gravity systems and highlight qualitative differences between integer- and half-integer-spin fields.