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Connecting the Electroweak Sphaleron with Gravitational Waves

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Connecting the Electroweak Sphaleron with Gravitational Waves PHYSICAL REVIEW D 101, 091903(R) (2020) Rapid Communications Connecting the electroweak sphaleron with gravitational waves † Ruiyu Zhou,1 Ligong Bian ,1,* and Huai-Ke Guo2, 1Department of Physics, Chongqing University, Chongqing 401331, China 2Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019, USA (Received 16 October 2019; revised manuscript received 2 February 2020; accepted 11 May 2020; published 21 May 2020) We study and reveal the relation between the electroweak sphaleron energy and the gravitational wave signals from a first-order electroweak phase transition. For the first time, we find that a larger sphaleron energy corresponds to a stronger gravitational wave signal for a sufficiently strong phase transition. We also present the correlation between the sphaleron and the properties of the gravitational waves, such as the signal-to-noise ratio, peak frequency, and amplitude, with the intention that traces of the sphaleron can be obtained with detailed measurement and studies of the stochastic gravitational wave at future space-based detectors. This study paves the way for a more practical revelation of the sphaleron which is otherwise unprobable at high energy colliders. DOI: 10.1103/PhysRevD.101.091903 I. INTRODUCTION sphaleron, which is otherwise inaccessible at colliders, correlates with the properties of the stochastic gravitational The first direct detection of gravitational wave signals waves and can be traced through future measurements and from the binary black hole merger by LIGO [1] and the studies of the stochastic gravitational waves. approval of the space-based interferometer LISA [2] have We first summarize the relations among the phase raised growing interest on the study of gravitational waves transition strength, gravitational waves from the EWPT, from a first-order electroweak phase transition (EWPT) in and the baryon number preservation criterion (BNPC) here: the early universe. It opens an alternative novel approach (1) a higher detectability of the stochastic background of to study new physics because the nature of the phase gravitational waves generally requires a smaller β=H transition in the standard model of particle physics is a n – (roughly the inverse time duration of the phase transition) crossover [3 6]. A strongly first-order EWPT provides a and a larger α (the latent heat normalized by the radiation nonequilibrium environment for baryon number generation energy) of the EWPT, where the sound waves in the plasma and fulfills one of the three Sakharov conditions [7], with dominates the gravitational wave production, and this which it is possible to generate the observed baryon corresponds to a higher strength of the EWPT [12,13]. asymmetry of the Universe [8] in the framework of the Both parameters are highly related with the finite temper- electroweak baryogenesis (EWB). On the other hand, ature potential that determines the sphaleron energy the (B þ L)-violating sphaleron process associated with (EsphðTÞ) inside the electroweak bubbles. (2) The spha- the change of Chern-Simons numbers [9,10] should be leron rate in the broken phase is proportional to a highly suppressed to avoid the washout of the baryon Boltzmann factor Γ ∝ exp½−E ðTÞ=T [11,14], implying asymmetry inside the electroweak bubbles where electro- sph that the estimation of E ðTÞ is crucial to determine if the weak symmetry is broken [11]. The behavior of the sph electroweak sphaleron process is sufficiently quenched electroweak sphaleron is directly correlated with the inside the electroweak bubbles to keep the baryon asym- possible patterns of EWPT and the generated gravitational metry generated during EWPT. Quantitatively, this means waves since the sphaleron energy is closely connected with the sphaleron rate in the broken phase should be lower than the phase transition order parameters at finite temperature the Hubble expansion rate, serving as the definition of the as will be shown in this paper. Therefore, the electroweak BNPC [8,15]. (3) The conventionally adopted strongly first-order EWPT condition [16](vðTÞ=T ≳ 1) is usually *[email protected] obtained after taking into account the scaling law [17], † [email protected] which connects the sphaleron energy at the temperature of ð Þ phase transition (Esph T ) to that at the zero temperature Published by the American Physical Society under the terms of (E )as[18,19] E ðTÞ ≈ E vðTÞ=v, where vðTÞ and v the Creative Commons Attribution 4.0 International license. sph sph sph Further distribution of this work must maintain attribution to are, respectively, the phase transition order parameter and the author(s) and the published article’s title, journal citation, Higgs vacuum expectation value (vev) at zero temperature. 3 ≈ 9 and DOI. Funded by SCOAP . Here Esph TeV, and thus leads to highly rare events for 2470-0010=2020=101(9)=091903(7) 091903-1 Published by the American Physical Society RUIYU ZHOU, LIGONG BIAN, and HUAI-KE GUO PHYS. REV. D 101, 091903 (2020) sphaleron-induced (B þ L)-violating process at the high extended Higgs sector. A simplified benchmark model is energy colliders in the current and near future [14,20–23] the gauge singlet extension of the SM, the xSM [30–32], and makes its detection unlikely at colliders [24]. with the finite temperature potential being [13] Based on the above relations, we revisit the relation 1 1 between BNPC and the strongly first-order EWPT cri- 2 2 2 V½h;s;T¼− ½m −Π ½Th − ½−b2 −Π ½Ts terion. We first calculate the electroweak sphaleron energy 2 h 2 s ð Þ ð Þ Esph T and the phase transition strength v T =T at the 1 4 1 2 1 2 2 b3 3 b4 4 þ λh þ a1h sþ a2h s þ s þ s ; ð2Þ temperature of the EWPT. After that, we check if the BNPC 4 4 4 3 4 is met with the following condition being satisfied [17]: where ΠhðTÞ and ΠsðTÞ are the thermal masses of the E ðTÞ ð Þ fields, ≡ sph − 7 v T þ T PTsph ln ln 100 T T GeV 2 2 2 2m þ m þ 2m λ a2 > ð35.9–42.8Þ; ð1Þ Π ½T¼ W Z t þ þ T2; h 4v2 2 24 The numerical range here originates from the uncertainty a2 b4 2 −4 −1 Π ½T¼ þ T : ð3Þ calculating the fluctuation determinant κ ¼ð10 –10 Þ s 6 4 [25], which is comparable to the uncertainty associated with the numerical lattice simulation of the sphalerons at the The scalar cubic terms in Eq. (2) dominate the phase standard model electroweak crossover [4]. The lattice sim- transition dynamics and can trigger a first-order EWPT ulation of the sphaleron rate can settle down the exact value after theoretical and experimental bounds on model param- of the right-hand side in Eq. (1) [26,27], which however is eters are taken into account [13]. Moreover, in this work, currently absent in the literature. Fortunately, our findings in we focus on the one-step EWPT with the electroweak this paper only mildly rely on this uncertainty. We thus vacuum denoted by (≡ðv; vsÞ). While two-step EWPT can investigate the BNPC condition and the scaling law, and its also exist, it is of negligible parameter space here. For the relation with the phase transition strength. Our study shows phase transition study, we do not take into account the CP 2 that the scaling law can be established when PTsph ∼ Oð10 Þ violation since it is generally very constrained by electric for the EWPT that satisfies the BNPC condition. A strongly dipole moment limits [33]. first-order EWPT that can produce detectable gravitational From an effective field theory point of view, a first-order waves generally is accompanied by a large sphaleron energy EWPT can be realized by inclusion of higher dimensional as Esph ∝ vn, as will be explored in this work. Based on that, operators, irrespective of a specific scenario. Among the we build the connection between the sphaleron energy and dimension-six operators of the SMEFT, the operator O6 the signal-to-noise ratio of resulting stochastic gravitational dominates the contribution to the Higgs potential and leads waves from the EWPT. to the following finite temperature effective potential: There are mainly two classes of new physics that can 1 λ h6 enhance the strength of the EWPT [8]: (1) one class couples V ðh; TÞ¼− ðm2 − c Þh2 þ h4 þ ; ð4Þ the standard model Higgs to new scalars and enhances the T 2 hT 4 8Λ2 cubic terms in the thermal potential through loop effects, ¼ð4 2 þ 3 2 þ 02 þ 8λÞ 2 16 with the minimal supersymmetry standard model as the where chT yt g g T = , and it is the celebrated example; and (2) the other couples the SM Higgs presence of the last term that allows the EWPT to be first to new scalar that develops a vev near the electroweak order through modifying the cubic terms of the potential at scale. The first class is less prioritized due to the current tree level in comparison with the SM case [34,35]. The severe theoretical or experimental constraints [5,28], and first-order EWPT parameter spaces with deviation of the we thus consider the second class where the first-order Higgs trilinear coupling from the SM can be tested by phase transition is triggered by tree-level cubic terms. We future colliders [36,37]. The requirement of the electro- use the extensively studied singlet extended standard model weak minimum being the global one results in the condition Λ ≥ 2 (“xSM”) and the standard model effective field theory v =mh, and the EWPT can be firstpffiffiffi order when the 2 (SMEFT) as two representative examples [29], with the potential barrier can be raised with Λ < 3v =mh [34,38].
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