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

This is the authors' abstract. We don't add key points for this paper.

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.