Extremal Higgs couplings

Abstract

We critically assess to what extent it makes sense to bound the Wilson coefficients of dimension-six operators. In the context of Higgs physics, we establish that a closely related observable, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>c</a:mi><a:mi>H</a:mi></a:msub></a:math>, is well defined and satisfies a two-sided bound. <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>c</c:mi><c:mi>H</c:mi></c:msub></c:math> is derived from the low momentum expansion of the scattering amplitude, or the derivative of the amplitude at the origin with respect to the Mandelstam variable <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>s</e:mi></e:math>, expressed as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>M</g:mi><g:mo stretchy="false">(</g:mo><g:msub><g:mi>H</g:mi><g:mi>i</g:mi></g:msub><g:msub><g:mi>H</g:mi><g:mi>i</g:mi></g:msub><g:mo stretchy="false">→</g:mo><g:msub><g:mi>H</g:mi><g:mi>j</g:mi></g:msub><g:msub><g:mi>H</g:mi><g:mi>j</g:mi></g:msub><g:mo stretchy="false">)</g:mo><g:mo>=</g:mo><g:msub><g:mi>c</g:mi><g:mi>H</g:mi></g:msub><g:mi>s</g:mi><g:mo>+</g:mo><g:mi>O</g:mi><g:mo stretchy="false">(</g:mo><g:msub><g:mi>g</g:mi><g:mi>SM</g:mi></g:msub><g:mo>,</g:mo><g:msup><g:mi>s</g:mi><g:mrow><g:mo>−</g:mo><g:mn>2</g:mn></g:mrow></g:msup><g:mo stretchy="false">)</g:mo></g:math> where <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi>g</n:mi><n:mi>SM</n:mi></n:msub></n:math> represents all Standard Model couplings. This observable is and, as a result, not sign-definite. We also determine the conditions under which the bound on <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mi>c</p:mi><p:mi>H</p:mi></p:msub></p:math> is equivalent to a bound on the dimension-six operator <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi>O</r:mi><r:mi>H</r:mi></r:msub><r:mo>=</r:mo><r:mo>∂</r:mo><r:mo stretchy="false">|</r:mo><r:mi>H</r:mi><r:msup><r:mo stretchy="false">|</r:mo><r:mn>2</r:mn></r:msup><r:mo>∂</r:mo><r:mo stretchy="false">|</r:mo><r:mi>H</r:mi><r:msup><r:mo stretchy="false">|</r:mo><r:mn>2</r:mn></r:msup></r:math>. Published by the American Physical Society 2024

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