Nonperturbative determination of the sphaleron rate for first-order phase transitions

In many extensions of the Standard Model electroweak phase transitions at high temperatures can be described in a minimal dimensionally reduced effective theory with SU(2) gauge field and fundamental Higgs scalar. In this effective theory, all thermodynamic information is governed by two dimensionless ratios <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>x</a:mi> <a:mo>≡</a:mo> <a:msub> <a:mi>λ</a:mi> <a:mn>3</a:mn> </a:msub> <a:mo>/</a:mo> <a:msubsup> <a:mi>g</a:mi> <a:mn>3</a:mn> <a:mn>2</a:mn> </a:msubsup> </a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>y</c:mi> <c:mo>≡</c:mo> <c:msubsup> <c:mi>m</c:mi> <c:mn>3</c:mn> <c:mn>2</c:mn> </c:msubsup> <c:mo>/</c:mo> <c:msubsup> <c:mi>g</c:mi> <c:mn>3</c:mn> <c:mn>4</c:mn> </c:msubsup> </c:math> , where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>λ</e:mi> <e:mn>3</e:mn> </e:msub> </e:math> , <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msubsup> <g:mi>m</g:mi> <g:mn>3</g:mn> <g:mn>2</g:mn> </g:msubsup> </g:math> , and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>g</i:mi> <i:mn>3</i:mn> </i:msub> </i:math> are the effective thermal scalar self-interaction coupling, the thermal mass and the effective gauge-coupling, respectively. By using nonperturbative lattice simulations to determine the rate of sphaleron transitions in the entire <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mo stretchy="false">(</k:mo> <k:mi>x</k:mi> <k:mo>,</k:mo> <k:mi>y</k:mi> <k:mo stretchy="false">)</k:mo> </k:math> -plane corresponding to the Higgs phase, and by applying previous lattice results for the bubble nucleation, we find a condition <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>x</o:mi> <o:mo stretchy="false">(</o:mo> <o:msub> <o:mi>T</o:mi> <o:mi>c</o:mi> </o:msub> <o:mo stretchy="false">)</o:mo> <o:mo>≲</o:mo> <o:mn>0.025</o:mn> </o:math> to guarantee preservation of the baryon asymmetry, which translates to <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>v</s:mi> <s:mo>/</s:mo> <s:msub> <s:mi>T</s:mi> <s:mi>c</s:mi> </s:msub> <s:mo>≡</s:mo> <s:msqrt> <s:mrow> <s:mn>2</s:mn> <s:mi mathvariant="normal">Δ</s:mi> <s:mo stretchy="false">⟨</s:mo> <s:msup> <s:mi>ϕ</s:mi> <s:mi>†</s:mi> </s:msup> <s:mi>ϕ</s:mi> <s:mo stretchy="false">⟩</s:mo> </s:mrow> </s:msqrt> <s:mo>/</s:mo> <s:msub> <s:mi>T</s:mi> <s:mi>c</s:mi> </s:msub> <s:mo>≳</s:mo> <s:mn>1.33</s:mn> </s:math> for the (gauge-invariant) discontinuity in Higgs condensate. This indicates that viability of the electroweak baryogenesis requires the phase transition to be slightly stronger than previously anticipated. Finally, we present a general template for analysing such viability in a wide class of beyond the Standard Model theories, in which new fields are heavy enough to be integrated out at high temperature.