Complete set of positivity constraints on the HEFT at NLO

We present the complete set of positivity bounds on the Higgs effective field theory (HEFT) at next-to-leading order (NLO). We identify the 15 operators that can be constrained by positivity, as they contribute to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>s</a:mi> <a:mn>2</a:mn> </a:msup> </a:math> -growth in the amplitude for longitudinal gauge-Higgs scattering, that is to all possible 2-to-2 scattering processes involving longitudinal gauge bosons, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>V</c:mi> <c:mi>L</c:mi> </c:msub> <c:mo>=</c:mo> <c:msubsup> <c:mi>W</c:mi> <c:mi>L</c:mi> <c:mo>±</c:mo> </c:msubsup> <c:mo>,</c:mo> <c:msub> <c:mi>Z</c:mi> <c:mi>L</c:mi> </c:msub> </c:math> , and the Higgs boson, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>h</e:mi> </e:math> . We find two sets of constraints: (i) specific linear combinations of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>C</g:mi> <g:mi>P</g:mi> </g:math> -even Wilson coefficients (WCs) must be positive, and (ii) the magnitudes of some WCs—including all <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>C</i:mi> <i:mi>P</i:mi> </i:math> -odd ones—must be smaller than products of other <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>C</k:mi> <k:mi>P</k:mi> </k:math> -even WCs. We present our final constraints on the 15 dimensional HEFT space and show how known positivity bounds on the 3 dimensional space of dimension 8 SMEFT can be recovered from them. We find that only about 5% of the parameter space for WCs of HEFT operators at NLO complies with these positivity constraints. Additionally, we obtain double-sided bounds on these WCs by fully exploiting the implications of unitarity and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>s</m:mi> <m:mi>t</m:mi> </m:math> -crossing symmetry. For WCs contributing to the vector boson scattering process our final constraints are in most cases significantly stronger than the experimental ones. For the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>V</o:mi> <o:mi>L</o:mi> </o:msub> <o:msub> <o:mi>V</o:mi> <o:mi>L</o:mi> </o:msub> <o:mo>,</o:mo> <o:mi>h</o:mi> <o:mi>h</o:mi> <o:mo stretchy="false">→</o:mo> <o:mi>h</o:mi> <o:mi>h</o:mi> </o:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>V</r:mi> <r:mi>L</r:mi> </r:msub> <r:msub> <r:mi>V</r:mi> <r:mi>L</r:mi> </r:msub> <r:mo>,</r:mo> <r:mi>h</r:mi> <r:mi>h</r:mi> <r:mo stretchy="false">→</r:mo> <r:msub> <r:mi>V</r:mi> <r:mi>L</r:mi> </r:msub> <r:mi>h</r:mi> </r:math> process, there are no reported experimental limits and our theoretical constraints provide the first bounds.