Electroweak Baryogenesis from Temperature-Varying Couplings
UUCI-HEP-TR-2019-13
Electroweak Baryogenesis from Temperature-VaryingCouplings
Sebastian A. R. Ellis a Seyda Ipek b and Graham White c a SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA b Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 USA c TRIUMF Theory Group, 4004 Wesbrook Mall, Vancouver, B.C. V6T2A3, Canada
E-mail: [email protected] , [email protected] , [email protected] Abstract:
The fundamental couplings of the Standard Model are known to vary as afunction of energy scale through the Renormalisation Group (RG), and have been measuredat the electroweak scale at colliders. However, the variation of the couplings as a functionof temperature need not be the same, raising the possibility that couplings in the earlyuniverse were not at the values predicted by RG evolution. We study how such temperature-variance of fundamental couplings can aid the production of a baryon asymmetry in theuniverse through electroweak baryogenesis. We do so in the context of the Standard Modelaugmented by higher-dimensional operators up to dimension 6. a r X i v : . [ h e p - ph ] M a y ontents α B.1 Step 1: Phase Transition 26B.2 Transport Equations 28
The Standard Model (SM) of particle physics provides an elegant explanation for manyof the observable phenomena in the universe. However, it fails to explain a few criti-cal phenomena, including one which is crucial to our existence, namely the origin of thematter-antimatter asymmetry in the universe. Through cosmological observations fromthe Cosmic Microwave Background (CMB) [1] and Big Bang Nucleosynthesis (BBN) [2],this asymmetry is measured to be Y B = n B s ∼ − . (1.1)The SM fails to explain the origin of the baryon asymmetry of the universe (BAU) becauseit does not satisfy two of the three Sakharov conditions [3]. While the SM has SU (2) L – 1 –phalerons which conserve B − L but violate B + L , it does not have enough CP -violation,nor does it contain the necessary out-of-equilibrium process. This failure of the otherwiseextremely successful SM is a compelling reason to demand new physics beyond the StandardModel (BSM).BSM models that address the problem of generating the BAU often include new fieldsthat couple to the Higgs field such that the electroweak (EW) transition becomes a first-order phase transition, which provides the out-of-equilibrium condition. In addition, thesenew couplings can be the source of the additional CP violation needed to explain the BAU.(For recent reviews, see [4, 5].)Both the CP violation and the out-of-equilibrium processes involved in the generationof the BAU depend on the gauge and Yukawa couplings at the time of the EW transition.For example, it has been shown that by modifying Yukawa couplings in the early universe, CP violation in the SM can be enhanced [6–10]. It has also been shown that gauge couplingsin the early universe can be different from their expected values given renormalisation grouprunning in the SM [11].In this work we investigate how changes in weak and strong coupling constants duringthe EW transition affect EW baryogenesis scenarios. To be more specific, we use anEffective Field Theory (EFT) framework which encapsulates the interactions of light fieldswith decoupled new physics. This formalism allows us to introduce the operators requiredto satisfy the Sakharov conditions, as well as modify the SM gauge couplings. The effectof operators required to ensure a first order phase transition and provide new CP violationare well understood, and have been studied extensively in the literature [12–17]. The novelaspect of our work is to include dimension-5 operators involving new scalar fields, whichchange the effective gauge couplings in the early universe.The effect of modified gauge couplings can be separated into two categories. Themost intuitive category is how changing the SU (2) L and SU (3) c coupling constants, α and α , directly affect the rates of the EW and strong sphalerons respectively. Sincesphaleron rates for an SU ( N ) symmetry group scale as α N , it is clear that even modestvariations in the coupling constants can result in large variations in the rates. Since EWbaryogenesis relies on out-of-equilibrium SU (2) L sphaleron processes to generate a baryonasymmetry, one would expect that making these processes more efficient should enhancethe production of baryons, as long as the same enhanced sphalerons do not subsequentlyerase the produced asymmetry. Additionally, EW sphaleron transitions are more effectivein the presence of a chiral asymmetry (sometimes called a CP asymmetry), generated byinteractions on the wall of the expanding bubbles of true vacuum. This chiral asymmetryis efficiently washed out by strong sphalerons, so that reducing their strength should alsohelp with generating the observed baryon asymmetry. The less intuitive category of effectsis how changing gauge couplings affects the finite-temperature Higgs potential, and thecalculation of the transport of particles in the vicinity of the true vacuum bubble wall.These effects are mostly due to modified thermal masses of the relevant gauge bosons andaltered interaction rates.We perform a calculation of these effects on the baryon asymmetry, as well as presentmodels which could give rise to the variation in the gauge couplings. We do so in a model– 2 –hich would otherwise be ruled out by current electron electric dipole moment (EDM)constraints, demonstrating the power of such variations in the gauge couplings.In Section 2 we present the EFT model we will use for our analysis. In Section 3 wediscuss the EW phase transition, and the possible effects of modifying gauge couplings.In Section 4 we discuss in greater detail how we compute the baryon asymmetry, startingfrom the requirement of new sources of CP violation, continuing with a discussion of howthe modified gauge couplings affect the computation of the obtained baryon asymmetry,and ending with a discussion of how washout from enhanced SU (2) L sphalerons in thebroken phase of EW symmetry can be avoided. This section, and in particular Figures 3and 4, constitute the main results of our paper. In Section 5 we propose models whichcould give rise to the required modifications of gauge couplings in the early universe, whichare consistent with current measurements. Finally, in Section 6 we present our concludingremarks. In this section we describe the general features of the model(s) we work with. The failureof the SM to reproduce the observed baryon asymmetry requires the presence of new fieldsand interactions. When there is a separation of scales between heavy and light fields,the former can be integrated out. The effect of the heavy fields is then encapsulated inthe Wilson coefficients of higher dimensional operators involving only the remaining lightfields, suppressed by the heavy field scale. The SM augmented by such higher dimensionaloperators involving only SM fields is commonly referred to as the SM Effective Field Theory(SMEFT). In our analysis we will invoke additional fields that are near or below the EWscale, so that it is a more general EFT framework. Current constraints on the scale of newphysics from the LHC can be enough to warrant the use of an EFT approach to study theeffects of decoupled new physics. Additionally, in the case of understanding mechanismsof baryogenesis, the use of an EFT is further validated by the strong constraints on new CP violation beyond the SM, coming from the non-observation of neutron and electronEDMs. Together these can constrain new physics up to the PeV scale in some models.In this analysis, we consider a minimal set of higher dimensional operators that needto be added to the SM to ensure successful baryogenesis in the context of varying α and α in the early universe.The EW phase transition in the SM is a crossover, meaning that the Sakharov condi-tion of out-of-equilibrium processes is not satisfied. It has been shown that a dimension-6Higgs operator is sufficient to turn this crossover into a first-order phase transition, therebysatisfying one of the Sakharov conditions [12, 13, 16]. Therefore, (1) we include the oper-ator O = | H | , where H is the SU (2) L Higgs doublet field, which contains the physicalHiggs field h .The SM has CP violation in the quark sector, but it is much too small to ensuresuccessful baryogenesis [20]. To compensate for this, in SMEFT, new CP violation through There are also strong constraints on new sources of CP violation from quark flavour observables (seee.g. [18, 19]), but these involve flavour-changing operators, which we do not consider here. – 3 –ffective Higgs-fermion operators are added. It is expected that the top-Higgs operatoris the most important due to the large top Yukawa. However, it has been shown thatconstraints on this term rule it out for generating the BAU [14]. We will show later that thisoperator can generate enough CP violation while being consistent with EDM measurementswhen we allow for gauge couplings to vary during the EW transition. Hence (2) we addthe top-Higgs operator, O tH = ¯ Q L ˜ Ht R | H | , with ˜ H = iσ H ∗ , to our Lagrangian.Finally, (3) we add dimension-5 operators parameterising interactions between newscalar singlets and the EW and strong gauge kinetic terms, which will enable the variationof α and α in the early universe through the dynamics of these scalar fields.Given these three additions to the SM, we may write the following Lagrangian for themodel under consideration L ⊃ − g Y (cid:18) − c g Y ϕ Y Λ Y (cid:19) B µν B µν − g (cid:18) − c g ϕ Λ (cid:19) W µν,a W aµν − g (cid:18) − c g ϕ Λ (cid:19) G µν,A G Aµν − V SM ( H ) − c Λ | H | + δ CP V Λ ¯ Q L ˜ Ht R | H | , (2.1)where V SM ( H ) = µ H | H | + λ H | H | is the SM Higgs potential and Λ X are various newphysics scales, presumably unrelated to each other. At first, we will be agnostic regardinghow these higher dimensional operators are generated. In Section 5 we propose two UVcompletions for the dimension-5 operators, both of which will have testable properties.The first three terms in Equation (2.1) are crucial elements of this work. These termsensure that the effective weak and strong couplings become field-dependent,1 g i, eff ≡ g i (cid:18) − c g i ϕ i Λ i (cid:19) . (2.2)Therefore, if the scalar fields ϕ i have non-trivial dynamics in the early universe, this willlead to a variation of the gauge couplings at early times/large temperatures that does notimmediately follow from the Renormalisation Group equations. We will eventually considerUV completions for these dimension-5 operators which will enable us to change the valuesof the gauge couplings at temperatures around the EW phase transition. The change ingauge couplings will be parameterised henceforth by δα i = ( α i, eff − α i, SM ) | T = T EW . (2.3)In the following sections we investigate how this change in gauge couplings affects the EWphase transition, the resulting gravitational wave spectrum, and most importantly, thegeneration of the observed baryon asymmetry. The strong and weak coupling constants feed into the calculation of the BAU in EW baryo-genesis (EWBG) scenarios in several different ways and it is not always straightforwardto disentangle these contributions. In this section we describe how the order of the phase– 4 –ransition changes with varying the weak coupling from its SM value while remaining ag-nostic to the origin of this deviation. Note that the strong coupling constant does notplay a role in the EW phase transition. However, variations in α will be important forgenerating the BAU, as will be discussed in the next section.At temperatures higher than the EW scale ( T (cid:38)
100 GeV), the Higgs potential getsfinite-temperature corrections due to the interactions of the Higgs field, h , with particlesinside the plasma. The finite temperature Higgs potential is calculated extensively in theliterature. The effective potential can be written as a sum of zero-temperature and finite-temperature pieces, V ( h, T ) = V ( h ) + V CW ( h ) + V T ( h, T ) , (3.1)where the first term is the tree-level potential and middle term is the zero-temperatureColeman-Weinberg loop correction. Loop corrections include all particles that interact withthe Higgs. However, it is enough to focus on the ones with largest couplings, i.e. Higgsself-coupling, gauge bosons and the top quark. For a given value of Λ / √ c , the parametersin the effective potential are fixed by requiring that the zero-temperature Higgs mass andvacuum expectation value (vev) are given at 1-loop by their experimentally measured valuesfor a renormalization scale of µ = m Z . At finite temperature the renormalization scale is setto the temperature µ = T . The finite temperature corrections to the tree-level potentialare V T = (cid:88) i ∈ bosons n i T π J B (cid:20) m i + Π i T (cid:21) + (cid:88) i ∈ fermions n i T π J F (cid:20) m i T (cid:21) . (3.2)In the above the Π i terms are Debye masses that result from a resummation and areincluded to prevent the breakdown of perturbation theory, see, e.g. , [22–24].In the SM, setting c = 0 in Equation (2.1), the finite-temperature potential has asimple form V ( h, T ) ∼ D ( T − T ) h − ET h + λ h , (3.3)where D = 2 M W + M Z + 2 m t v , E = 2 M W + M Z πv , T = m h D , and v = 246 GeV is the zero-temperature Higgs vev. As can be seen from the aboveequation, the Higgs potential at high temperatures is dominated by the quadratic termand as such the EW symmetry is not broken. As the universe cools down, the potentialacquires a second minimum at non-zero (cid:104) h (cid:105) . At a critical temperature T c , this minimumbecomes the global minimum and the Higgs field has a non-zero vev v c .The details of this transition are crucial for generating the BAU. Specifically, in order tosatisfy the out-of-equilibrium condition, a first-order phase transition (FOPT) is required.The strength of the phase transition is measured by comparing v c and the the size of the Note that the analysis of the thermal parameters has some renormalization dependence as shown in[21]. We find that this effect gets accentuated near the limits of our parameters space, i.e. δα ∼ . – 5 – .00 0.02 0.04 0.06 0.08 0.101.41.61.82.02.22.42.6 Figure 1 : Ratio of the Higgs vev at the critical temperature T c to the critical temper-ature as a function of δα for different values of the dimension-6 operator, Λ / √ c =575 , ,
675 GeV. The dashed line shows the required v c /T c for a first-order phase transi-tion given Λ / √ c = 600 GeV. Changing Λ / √ c does not affect this dashed line substan-tially. For details see Section 4.2.barrier that separates the false vacuum at (cid:104) h (cid:105) = 0 and the true vacuum at (cid:104) h (cid:105) = v c . In theSM and in most BSM scenarios, this condition translates into v c /T c (cid:38)
1. (In Section 4 wewill show how this condition is modified in our model.) From Equation (3.3), and taking g → g + δg at T c , we get v c T c ∼ Eλ (cid:12)(cid:12)(cid:12)(cid:12) T = T c ∼ m h × ( g + δg ) g ∼ . g + δg ) g , (3.4)where we ignore variations in g Y . It can be seen that the condition for a FOPT is notsatisfied in the SM. From the above expression, it is clear that in order to get a FOPT,the EW coupling must be considerably modified. Throughout our paper we will refer to v c /T c as the measure of the strength of the FOPT, as opposed to v n /T n , the correspondingquantities when bubbles of true vacuum begin to nucleate. Our use of v c /T c is somewhatconservative, as it is typically required to be larger than v n /T n .Alternatively, the barrier between the true and false vacua can be generated at treelevel via a non-renormalizable | H | operator in the potential [12], giving rise to a Higgsfield 6-point interaction, V = µ h − λh + c h . (3.5)This is what we do, as included in the Lagrangian in Equation (2.1). In this case v c islarger than T c for 538 GeV (cid:46) Λ √ c (cid:46)
800 GeV with the lower bound set by the requirement We numerically verified that the dependence on g Y is very weak. The obtained order parameter is actually smaller than the above scaling would suggest after one includesall 1-loop corrections to the Higgs potential, including the so-called “daisy” corrections. – 6 – .00 0.02 0.04 0.06 0.08 0.1051050100
Figure 2 : Shown here are contours where the correct baryon asymmetry is obtained asa function of δα and the scale of new CP violation Λ CPV / √ δ CPV , for different choices of δα and a fixed choice of Λ / √ c = 575 GeV. In light blue is where α is kept at its SMvalue, while the purple and pink contours are shown for δα = − . , − . CPV / √ δ CPV > . v c /T c changes while varying δα anddifferent values of Λ / √ c . It can be seen that one obtains a large v c /T c for small valuesof Λ / √ c , as expected. Raising the weak coupling constant raises v c /T c for small valuesof δα . Large variations in the weak coupling has the opposite effect. We suspect at largervalues of δα , contributions from daisy diagrams become more important than the leadingorder analysis, as in Equation (3.4).Before moving on to calculating the BAU, we also note that changing the weak cou-pling constant also changes the gravitational wave spectrum. expected from an EW phasetransition. We include a brief discussion of this in Appendix A. In this section we calculate the baryon asymmetry produced in a SMEFT scenario withvarying weak and strong coupling constants.
Electroweak baryogenesis with CP violation entering through the top Yukawa coupling andan | H | operator to induce a first order phase transition is ruled out by the latest electronEDM constraint [26], Λ CPV /δ CPV > . ashoutNo BAUNo Nucleation0.00 0.02 0.04 0.06 0.08 0.10550600650700 Figure 3 : Viable parameter space as a function of δα and Λ / √ c , for δα = 0. Thegrey region at lower values of Λ / √ c is ruled out due to bubbles of true vacuum notbeing able to nucleate. The transparent grey region on the left shows where an insufficientBAU is produced given Λ CPV / √ δ CPV = 7 . R , defined in Eq.Equation (4.14), of 1, 1 .
1, 1 . . CPV √ δ CPV = 7 . . (4.1)In Figure 2 we show how the required scale of CP violation changes by varying the weakand strong coupling constants.Altering the gauge couplings in the early universe can dramatically alter the efficiencyfor producing a baryon asymmetry by changing the weak and strong sphaleron rates. In afirst-order phase transition, bubbles of true vacuum (where the EW symmetry is broken)nucleate and grow. Outside of these bubbles, the EW symmetry remains unbroken. CP -violating interactions between the quarks and the bubble wall catalyze a chiral asymmetryin front of the bubble. Due to its large Yukawa coupling, the most important particleinvolved in these interactions is the top quark. This chiral asymmetry biases the EWsphalerons to create a baryon asymmetry in front of the bubble wall, some of which isswept up into the broken phase by the expanding bubble. Since the EW sphalerons areresponsible for generating the baryon asymmetry, enhancing them in the unbroken phasecan lead to a greater final baryon asymmetry. A detailed discussion of the effect of thisenhancement persisting in the broken phase can be found in Section 4.2. In order for this– 8 – ashoutNo BAU No Nucleation0.00 0.02 0.04 0.06 0.08 0.10550600650700 Figure 4 : Viable parameter space as a function of δα and Λ / √ c for δα = 0. The greyregion at lower values of Λ / √ c is ruled out due to bubbles of true vacuum not being ableto nucleate. The transparent grey region on the left shows where an insufficient BAU isproduced. The red region is where an initially sufficient BAU is produced, but the phasetransition is too weakly first order, so that the EW sphalerons wash out the BAU in thebroken phase, assuming a washout factor of F W = 10%, defined in Eq. Equation (4.6).The light and dark green contours correspond to obtaining the correct BAU, and a factorof ten more than required, respectively. Also shown are contours corresponding to the 68%CL bounds from HL-LHC (dashed) and ILC-250 (dot-dashed), as found in [25].process to be efficient, the chiral asymmetry produced in front of the bubble wall shouldnot be washed out by strong sphalerons. Thus, while increasing the EW sphaleron ratein the unbroken phase would increase the baryon asymmetry, likewise a suppression of thestrong sphaleron rate would lead to less washout of the chiral asymmetry, and therefore agreater baryon asymmetry.In order to get the most accurate results, sphaleron/instanton rates should be cal-culated using non-perturbative techniques such as on a lattice. However analytical ap-proximations exist for both the weak [27] and the strong sphaleron rates [28] in thermalequilibrium and describe the underlying phenomena to a good degree,Γ WS (cid:39) α T and Γ SS (cid:39) α T . (4.2)The weak sphaleron rate is given here in the symmetric phase, and is suppressed by an ex-ponential factor ∼ exp( − M W /T ) in the broken phase. Equation (4.2) shows the sensitivityof the strong and weak sphaleron rates to the gauge couplings. We emphasize that the maineffect of modifying the gauge couplings is to change these sphaleron rates. While we applythis variation of gauge couplings to a particular model in this paper, the inclusion of therelevant dimension-5 operators of Equation (2.1) in any other model of EW baryogenesiscan also be investigated. We leave such studies to future work.– 9 –n computing the final baryon asymmetry, we start by finding the profile of the truevacuum bubble. This step is required because we invoke a non-renormalizable operator asthe source of CP violation, so that the baryon asymmetry is sensitive to the bubble wallwidth [15]. We obtain the bubble wall profile by solving the classical equations of motionof the Higgs field across the spatial boundary between regions of true and false vacuum.This solution is a smooth function, and can be fit to a tanh( x ) function for ease in theremainder of the calculation. We then make use of the vev-insertion approach outlined in[29] to calculate the chiral asymmetry produced from CP -violating interactions with thebubble wall. We then calculate the resultant baryon asymmetry produced by the SU (2) L sphalerons as an integral over the left-handed fermion density n L [31, 32] Y B = 3Γ WS ( α )2 sD Q ( α i ) κ + ( α i ) (cid:90) −∞ d y n L ( α i , y ) e − κ − ( α i ) y . (4.3)The quantity κ ± is defined as κ ± = v w ± (cid:112) v w + 15 D Q ( α i )Γ WS ( α )2 D Q ( α i ) , (4.4)where v w is the wall velocity, D Q ( α i ) is the diffusion coefficient which depends on α i (seeEquation (B.14)), Γ WS is the SU (2) L sphaleron rate, and s is the entropy density. Adetailed discussion of how we compute Y B is given in Appendix B.From Equation (4.3), we can observe that increasing the weak sphaleron rate in theunbroken phase will increase the final baryon yield. The growth of Y B with α is quitedramatic since the weak sphaleron rate grow as α . However, the enhancement frommodified sphaleron rates will not diverge for the following two reasons. First, if we assumean exponential profile for n L ( z ) in Eq. Equation (4.3), a linear term in Γ WS exists in boththe numerator and denominator, so that for large enough Γ WS the yield Y B asymptotes.In addition, while initially increasing α raises the masses of the SU (2) L doublets m L suchthat CP -violating interactions with the bubble wall approach a resonance, eventually α isso large that m L > m R , which leads to non-resonant interactions (further details given inthe appendix around Equation (B.19). Finally, α modifies the bubble wall profile whichthereby changes the profile of the CP -violating source. More details of how modificationsof α enter every step of the calculation are provided in Appendix B.The dependence of the baryon asymmetry on α mostly arises from the suppression ofthe strong sphaleron rate. The strong sphaleron rate relaxes the chiral asymmetry. There-fore, if the strong sphaleron rate is decreased, then n L , the chiral asymmetry, increases.Again this effect is quite dramatic due to the α scaling of the strong sphaleron rate. Inaddition the reduction in the strong coupling increases the diffusion coefficient which tendsto moderately enhance the BAU. Finally this growth of the BAU is resisted slightly by the The accuracy of the vev-insertion approach is yet to be thoroughly tested and it remains unclear whetherit results in an under- or over-estimate of the baryon asymmetry. However, a more rigorous treatment ofa toy model did confirm the existence of a resonance-like feature [30]. Therefore, numerical values of thebaryon asymmetry we find should be understood as being indicative of the dependence of the BAU on theparameters of the theory, as opposed to a precise calculation of the final baryon abundance. – 10 – .00 0.02 0.04 0.06 0.08 0.10550600650700
Figure 5 : Contours of the sphaleron energy in units of 4 πv/g as a function of both δα and Λ / √ c , as defined in Equation (4.12).fact that the CP -violating sources are largest for m L = m R = 2 T and the thermal massesare decreasing with smaller α . We again relegate further details to Appendix B.We emphasize that the result of Equation (4.3) is the baryon asymmetry produced inthe unbroken phase and in front of the bubble wall, without accounting for dynamics in thebroken phase of EW symmetry. We discuss in the next section how this asymmetry canbe washed out if weak sphalerons are still active in the broken phase, inside the bubble. The success of the mechanism presented here depends strongly on the enhanced sphaleronrate in the unbroken phase of EW symmetry and across the bubble walls. It therefore alsodepends strongly on ensuring that the enhanced sphaleron rate, if it persists in the brokenphase inside the bubble, does not wash out the generated baryon asymmetry. In order forthis washout to be prevented, it is clear that the phase transition should be strongly firstorder. This would ensure that the change in the sphaleron rate between the unbroken andbroken phase is drastic, so that the enhanced sphaleron rate is nevertheless small insidethe bubble of true EW vacuum.Inside the bubble of true vacuum, baryon number density n B is depleted according to ∂ t n B = − N F sph ( T ) V T n B ∝ − e − E sph /T n B , (4.5)where E sph is the sphaleron energy. Given a phase transition duration t PT , we may definea washout factor as given by F W = n B ( t PT ) n B (0) , (4.6)which will provide a bound on the degree to which washout is tolerated in a model of baryo-genesis. This bound can be incorporated into an approximate baryon number preservation– 11 –ondition (BNPC) as a function of v c /T c [33], which is4 π E ( λ H , g , c ) g v c T c − v c T c > − log( − log F W ) − log t PT t H + log χ N + log κ , (4.7)where χ N = (cid:20)(cid:18) N F (cid:19) N tr ( N V ) rot (cid:18) ω − t H π (cid:19)(cid:21) , (4.8) t H = 1 /H ( T ) is the Hubble time, κ is the fluctuation determinant ratio that goes into cal-culating the EW sphaleron rate, ω − is the frequency of the unstable mode of the sphaleron, N F is the number of fermion families, and N tr ( N V ) rot ∼ κ is a sensitive function of λ H /g , and was computedin [35] for four values of λ H /g = 0 . , . , ,
10, from which an extrapolation in the range5 × − (cid:46) λ H /g ≤
10 was presented. In that analysis, it was found that for λ H /g ∼ . κ ∼ − .
2, while for λ H /g ∼ .
1, log κ ∼ − .
9. Since our modelincreases g , it pushes us to values of λ H /g < . κ isquite sensitive to the precise choice of δα . The largest deviation from the SM we willconsider is δα ∼ .
1, which corresponds to λ H /g ∼ .
08. This is at the lower limit ofthe interpolating curve shown in [35], which gives log κ ∼ −
11. Since an explicit numericalcalculation of the fluctuation determinant ratio as a function of δα is beyond the scope ofour study, and the calculation of [35] was only performed for four specific values, our choiceof log κ has an inherent uncertainty which we are currently unable to quantify. However,the perturbative calculation of log κ gives substantially larger values than the numericalresults of [35], so we can hope that the true value of log κ ( δα ) is not smaller than thatwhich we use, meaning that our estimate might be conservative.In our analysis of the washout avoidance condition, we will take − log F W ∼ min[0 . , log Y B /Y obs B ] , (4.9)which corresponds to a washout factor of either 10% or the amount by which our mechanismoverproduces a baryon asymmetry. We use this washout factor because we find that forlarge δα we can obtain a substantially larger BAU than is required, and therefore toleratea significantly larger washout factor.The function E ( λ H , g , c ) appearing in Equation (4.7) is the sphaleron energy in unitsof 4 πv/g . This function can be obtained only by computing the sphaleron solution of theeffective field theory by starting with the usual ansatz of [36] for the gauge and scalar fields W ai σ a = − ig f ( ξ ) ∂ i U U − , H = v √ h ( ξ ) U (cid:32) (cid:33) , (4.10)where U is an element of SU (2), and ξ = g vr is a dimensionless radial coordinate. Inthe presence of the c Λ | H | operator, the usual coupled non-linear differential equations of– 12 –otion for f ( ξ ) and h ( ξ ) are modified and can be written as [12] ξ d fdξ = 2 f (1 − f )(1 − f ) − ξ h (1 − f ) ,ddξ (cid:18) ξ dhdξ (cid:19) = 2 h (1 − f ) + λ H g ξ ( h − h + 34 v c g Λ ξ h ( h − . (4.11)We solve these differential equations numerically using the Newton-Kantorovich Method(NKM) as described in great detail in [37].The sphaleron energy at T = 0 is given by E sph = (cid:82) d xT , where T µν is the stress-energy tensor, and may be written as E sph = 4 πv E ( λ H , g , c ) g = 4 πvg (cid:90) ∞ dξ (cid:32) (cid:18) dfdξ (cid:19) + 8 ξ f (1 − f ) + 12 ξ (cid:18) dhdξ (cid:19) + h (1 − f ) + λ H g ξ ( h − + v c g Λ ξ ( h − (cid:33) . (4.12)In practice, we compute this integral numerically from the solutions h ( ξ ) and f ( ξ ) obtainedusing the NKM. We use a symmetric numerical integration procedure, where dXdξ = X i +1 − X i − ξ , (4.13)where X i = f i , h i corresponds to the value of the function at step i , and ∆ ξ is the separationbetween steps. The result for E ( λ H , g , c ) is shown in Figure 5. The behaviour is asexpected, in that at large δα , or correspondingly, small λ H /g , the normalised sphaleronenergy asymptotes towards a fixed value E ∼ . g → ∞ . For larger suppression scalesthan those shown in Figure 5, the sphaleron energy increases (because we have chosen c >
0) until we recover the SM Higgs sector in the limit Λ → ∞ [12, 37].The sphaleron energy is used in the calculation of the BNPC of Equation (4.7). Thisenables us to compute the required value of v c /T c that ensures that sphaleron rates aresufficiently suppressed inside the bubbles of true vacuum, so that our produced baryonasymmetry is not washed out. This requirement must then be contrasted with what isactually obtained in our model. We define the ratio R = v c /T c v c /T c | obt . | req . , (4.14)to parameterize the extent to which the baryon asymmetry produced by the enhancedout-of-equilibrium sphalerons is not washed out by the sphalerons remaining enhanced inthe broken phase. The values of v c /T c required range between 1 . (cid:46) v c /T c (cid:46) .
8, withthe larger values being required at large δα . Meanwhile the values of v c /T c obtained aretypically no larger than v c /T c ∼
2, and the largest values are obtained for low Λ / √ c andlow δα , as can be seen in Figure 1. – 13 – .3 Discussion of Results Due to the large number of moving parts in the calculation of the final baryon asymmetryobtained when varying gauge couplings, we discuss here briefly the main results, which areshown in Figures 2 to 4.In Figure 2, a particular choice of Λ / √ c = 575 GeV was made to show how the scaleof new CP violation that yields the correct baryon asymmetry varies as a function of δα .As mentioned, new CP violation in the top quark sector is ruled out by non-observationof the electron EDM if δα = δα = 0. This can be seen by the light blue contour onlyappearing for values of δα (cid:38) .
02. The location of these contours would shift downwardsfor higher values of Λ / √ c . All of this parameter space could potentially be probed infuture electron EDM searches [38–40].In Figure 3, we show the viable parameter space when we vary only δα and Λ / √ c ,taking δα = 0. Here, three effects compete to reduce the viable regions. On the onehand, if Λ / √ c is too low, the tunneling rate is too low, and bubbles of true vacuumcannot nucleate, as shown in the lower grey shaded region. On the other, the observedbaryon asymmetry cannot be reproduced if δα is too small, due to the constraint from theelectron EDM on the scale of new sources of CP violation. Finally, while a large baryonasymmetry can be produced by the enhanced sphalerons in a wide region of the parameterspace (above and to the right of the grey shaded regions), it is also washed out by the otherside of the double-edged sword that is a higher sphaleron rate. Indeed, as one increases δα ,the baryon asymmetry grows rapidly, and is often overproduced. However, if the sphaleronrate remains enhanced in the broken phase, for too large δα or Λ / √ c , the strength ofthe FOPT is insufficient to prevent washout of this initial overproduction. Thus, the viableparameter space is restricted to certain values of v c /T c which lie below the red contour of R = 1 (See Equation (4.14)).In Figure 4, we show the parameter space when varying only δα and Λ / √ c , given δα = 0. The region in which bubbles of true vacuum do not nucleate is determined byour choice of δα , such that for positive non-zero values of δα , this exclusion region wouldshift upwards. The region in which the initial BAU cannot be achieved would shift to theleft if δα > F W ∼
10% if δα were varied. This choice correspondsto a conservative estimate, since this constraint can in principle be relaxed if we were totake advantage of the initial overproduction of the baryon asymmetry.We do not present a combined analysis of simultaneous variations of α and α . Aswill be discussed in the next section, the UV completions envisaged to modify the gaugecouplings in the early universe tend to not apply simultaneously to both α and α .Interestingly, the vast majority of the open parameter space can be probed by con-straining the triple-Higgs coupling λ hhh at High Luminosity LHC (HL-LHC), and all of it(if only α is varied) could be probed at the ILC running at √ s = 250 GeV [25]. Shown inFigures 3 and 4 are 68% CL limits on Λ / √ c as obtained from a global fit, which for ILCassume a combination with HL-LHC. The 95% CL bounds for HL-LHC do not appear inthe figure shown. The proposals to run ILC at √ s = 250 , ,
500 GeV, or the FCC-ee– 14 –t √ s = 240 ,
350 combination could place 95% CL bounds on the entire parameter spaceshown in Figures 3 and 4.
In this section we discuss two different mechanisms for generating the dimension-5 oper-ators in the effective Lagrangian in Equation (2.1) which control the size of the variousgauge couplings at temperatures near that of the EW phase transition. We discuss howeach mechanism can only serve to modify one of either the weak or strong gauge couplingat a time, with experimental constraints preventing the simultaneous modification of theother. The first mechanism, which involves an ultra-light scalar with a non-zero energydensity which scan values of the gauge couplings, additionally requires modifying the hy-percharge gauge coupling so as to evade constraints during Big Bang Nucleosynthesis andfrom measurements of stars at O (1) redshifts. The second mechanism requires the exis-tence of an additional scalar field near the EW phase transition scale which obtains a vev.In order for this scalar to couple to gauge bosons sufficiently, a large number of new statesmust also exist near that scale, leading to possible experimental signatures. We first discuss the first mechanism, whereby an ultra-light scalar field, ϕ , scans values ofthe gauge couplings as it evolves in time. Such a scalar field may be treated as a coherentlyoscillating classical field whose energy density behaves like non-relativistic matter as longas its potential is dominated by the ϕ term [41]. This ultra-light scalar could be a lightmodulus or dilaton-like field, which we will henceforth treat as making up some fraction ofthe Dark Matter (DM) energy density ρ DM , denoted f DM . We therefore write the field as ϕ (cid:39) √ f DM ρ DM m ϕ cos ( m ϕ t ) , (5.1)where m ϕ is the mass of the ϕ field.If this field has couplings to the field strengths of the SM gauge groups, we may writethese as L ⊃ − g Y (cid:18) − c g Y ϕM Pl (cid:19) B µν B µν − g (cid:18) − c g ϕM Pl (cid:19) W a,µν W aµν − g (cid:18) c g g β ϕM Pl (cid:19) G A,µν G Aµν , (5.2)where B µν , W aµν , G Aµν are the U (1) Y , SU (2) L , SU (3) c field strength tensors respectivelyand we have set Λ Y, , in Equation (2.1) to a slightly modified value of the Planck mass, M Pl = (4 πG N ) − / = 3 . × GeV, to match the literature on dilaton couplings. Thenormalizations of the scalar couplings c g Y , c g , c g are also chosen to match the treatment Note that with our definition, M Pl = √ m P , where m P is the usual reduced Planck mass. – 15 –f dilaton couplings [42–44]. This choice of normalization requires a coupling of the scalarto fermion masses L ⊃ − c g ϕM Pl (cid:88) f γ m f m f ¯ ψ f ψ f , (5.3)such that the coupling c g appears in a RG-invariant manner. This formulation of thecoupling c g is such that it is equivalent to coupling ϕ to the anomalous part of the gluonstress-energy tensor, which in turn ensures that c g is a measure of the coupling of ϕ to thegluonic energy component of the mass of a hadron.These couplings of the ultra-light scalar to the gauge kinetic terms amount to a changein the effective gauge coupling, now given by g i, eff = g i (1 − c gi ϕM Pl ) / , (5.4)for the U (1) Y and SU (2) L couplings. The SU (3) c coupling modification is not written outbecause as we will discuss later, constraints on c g are such that no substantial modificationof α will be possible by this mechanism. Indeed, the constraint is such that only for m ϕ (cid:46) − eV can one achieve δα (cid:38) .
03 as required for successful baryogenesis in ourmodel. This value of m ϕ is barely compatible with the picture of a coherently oscillatingscalar field, which requires m ϕ (cid:38) H ∼ − eV, where H is the value of the Hubbleconstant today [45]. Since this is at the boundary of compatibility with constraints, wewill only consider modification of α Y and α in this discussion.If the scalar ϕ behaves as non-relativistic matter, its energy density will evolve as T from T eq ∼ ( m ϕ M Pl ) / to the temperature now, T , so that the field value of ϕ was( T /T ) / greater at a temperature T < T eq than it is now. Therefore, g i, eff will vary as afunction of the temperature between now and T eq . At temperatures above T eq , the value ofthe scalar ϕ behaves as dark energy would, and therefore does not change. In this section,we only consider scalars with masses below m ϕ (cid:28) − eV, i.e. T eq (cid:28) T EWPT , so thatduring the electroweak phase transition the scalar ϕ is not undergoing oscillations. The couplings of ultra-light scalars have been strongly constrained by searches for long-range Equivalence Principle (EP) preserving “fifth forces” and for EP-violating forces.They have also been strongly constrained by searches for variations in the fundamentalconstants.At low scalar masses, the strongest constraint on the coupling c g comes from a searchfor deviations in the gravitational lensing of the sun, performed by the Cassini spacecraft[46]. This search places a constraint on the Eddington parameter γ :1 − γ (cid:39) c g ≤ (2 . ± . × − , (5.5)such that | c g | (cid:46) × − at 2 σ . Hence any modifications of the SU (3) c coupling will beminimal, as claimed above. The RG-invariance holds only up to electromagnetic corrections which are α EM suppressed. – 16 –he searches for long-range forces are conducted at energies well below the EW sym-metry breaking scale, and so they typically refer to the coupling to the photon kinetic termas opposed to separating into the U (1) Y and SU (2) L kinetic terms. Thus they constrain acoupling c e , which appears in the Lagrangian as L ⊃ − e (cid:18) − c e ϕM Pl (cid:19) A µν A µν . (5.6)Using the relationship between electromagnetic (EM) coupling e and the U (1) Y and SU (2) L gauge couplings, e = g Y g ( g Y + g ) / , (5.7)we can express the coupling of the scalar to the EM kinetic term in terms of the couplings c g and c g Y as c e = α Y c g + α c g Y . (5.8)In turn, we may now discuss the constraints that have been set on variations of the EMfine structure constant α EM in terms of constraints on the scalar couplings to the U (1) Y and SU (2) L kinetic terms. The variation of the EM coupling as a function of the scalarfield value ϕ is (cid:18) ∆ α EM α EM (cid:19) = c e ϕM Pl − c e ϕ , (5.9)which will enable us to constrain c e , and in turn c g Y and c g .There exist strong constraints on the variation of the EM coupling from astrophysicaldata, taken at redshifts between 1 < z <
4. A meta-analysis of recent measurements yieldsa weighted mean [47] (cid:18) ∆ α EM α EM (cid:19) astro, ’17 = ( − . ± . × − , (5.10)which is consistent with zero at 1 σ , while a previous analysis [48] found (cid:18) ∆ α EM α EM (cid:19) astro, ’10 = ( − . ± . × − , (5.11)which indicated a possible variation of the EM coupling at high redshift, but remainedconsistent with zero at a little over 2 σ . The first average above for ∆ α EM /α EM correspondsto c e ∼ − ± × (cid:16) m ϕ eV (cid:17) f − / , (5.12)assuming a central observation redshift of z = 2 . ϕ interaction.Additionally, there is a strong constraint on the variation of the EM coupling from theOklo natural fission reactor [49], which constrains − . × − < (cid:18) ∆ α EM α EM (cid:19) Oklo < . × − , (5.13)– 17 –hich also constrains the annual variation of the EM coupling to be less than one part in ∼ . This constraint corresponds to a constraint − × (cid:46) c e (cid:16) m ϕ eV (cid:17) − f / (cid:46) × , (5.14)for a gravitationally-suppressed ϕ interaction as above.There are also constraints on the variation of α EM from Big Bang Nucleosynthesis(BBN), which can be of similar strength if the variation is coupled to variations of otherfundamental parameters [50]. However, since these coupled variations are not consideredin this paper, we will not discuss them further. There exist also constraints that can be set independently on m ϕ and f DM , comingfrom searches for EP-violating interactions. The strongest constraint was set by the E¨ot-Wash experiment [51], which compared the relative acceleration of Beryllium and Titaniumtest masses. Following the analysis of [43, 44], we find that this constrains the combinationof c e and c g to be (cid:12)(cid:12)(cid:12)(cid:12) − . × − c e − . × − c e c g + 6 . × − c g (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (0 . ± . × − . (5.15)We use this constraint in Figure 6 to show the upper limit on c e , and the re-interpretationas an upper limit on c g Y and c g in the absence of the other respectively. In the mass rangeof interest for ϕ , this, combined with the Cassini constraint on c g , provide the strongestconstraints.Given these constraints, we see that the maximum scalar coupling to the SU (2) L kinetic term in the absence of tuning is | c g | ∼ − if c g saturates the Cassini bound, or | c g | ∼ × − if | c g | = 0. This constraint is so strong that a change in δα would notbe possible, since the required scalar mass would be so small it would not be coherentlyoscillating now. These constraints might have been relaxed if a chameleon mechanism werepresent [52]. However, given the requirement that the ϕ term in the potential dominatesover potential cubic and quartic terms, the Cassini constraint will still apply [53].Crucially however, these constraints are not directly on c g , but instead are interpre-tations of constraints on c e on the former. If we can impose that c e = 0, then there wouldbe no existing constraint on c g explicitly derived. This can be achieved if we impose afine-tuning of the relation between c g Y and c g , namely c g Y c g = − α Y α , (5.16)which could have its origin in some symmetry. This relation corresponds to writing Equa-tion (5.2) as L ⊃ − g Y B µν B µν − g W a,µν W aµν − (cid:18) c g ϕg M Pl (cid:19) (cid:0) B µν B µν − W a,µν W aµν (cid:1) , (5.17) One might be concerned that BBN should constrain variations in α due to the sensitivity of the finalHelium abundance to the neutron lifetime, which is a weak process. The neutron decay rate can be writtenso that Γ n ∝ ( g /M W ) , so that it is sensitive to the Higgs VEV during BBN, but not g directly, at treelevel. – 18 – - - - - - - - - - Figure 6 : Constraints from EP-violation searches from the E¨ot-Wash experiment [51] on c e , c g Y and c g (blue, red, yellow) as a function of | c g | , which is allowed to vary up to itsmaximum of | c g | ∼ √ − as discussed in the text. The discontinuity at small values of | c g | corresponds to when the strongest constraint is on Im( c i ) as opposed to Re( c i ).with c g being set to zero. The B − W structure can arise, for example, from a left-right symmetric UV completion similar to that of [54], with ϕ coupling to W R − W L . Thisstructure can be postulated at a given scale µ , in which case a concern might be that unlessit is enforced by an unbroken symmetry, it will not be invariant under RG flow. We willnot consider this possibility here, and instead assume that the relation of Equation (5.16)holds in the IR.In principle, in the limit c e = 0, a constraint on c g can nevertheless be obtainedby computing the EW matrix element of the proton and neutron. Variation of α wouldthen lead to a small variation in the proton and neutron masses, which can be constrainedby the E¨ot-Wash result. We expect this constraint to be roughly α /α EM × GeV /m W weaker than the constraint on | c e | , which would translate into a bound | c g | (cid:46)
10 (0 .
8) for | c g | = 0 (6 × − ). A computation of the exact bound is left to future work. α Having established that c g can be non-zero, and indeed could be sizeable if c e is set tozero by the tuning condition of Equation (5.16), we now continue with the analysis of how α can be scanned by an ultra-light scalar.The shift in the weak structure constant can be written as δα = α c g ϕ/M Pl − c g ϕ/M Pl , (5.18)with ϕ given by Equation (5.1) such that δα (cid:39) α (cid:32) c g √ f DM . × (cid:0) m ϕ eV (cid:1) / − c g √ f DM (cid:33) , (5.19)– 19 – - - - - - - - - - Figure 7 : Left : Contours of constant δα for varying scalar mass and DM fraction f DM .The blue shaded region is excluded by CMB and large-scale structure (LSS) data [55],while the dashed black line corresponds α → ∞ , with α < c g = 1. Right : The variation in the weak coupling δα as a function ofthe mass for different choices of the scalar couplings c g . The DM fraction has been set to f DM = 0 .
025 to be free of CMB/LSS constraints. We allow large c g by enforcing c e = 0as discussed in the text.so that appreciable shifts in the weak structure constant will only occur for very smallscalar masses, in the vicinity of the pole at m ϕ ∼ − eV (cid:0) c g . (cid:1) (cid:16) f DM . (cid:17) . This meansthat if ϕ is to successfully scan α an appreciable amount, it must be ultra-light.Given this expected range for the scalar mass, one must ensure that the ultra-lightscalar is nevertheless sufficiently heavy that it is coherently oscillating at the present time.As discussed previously, this requires that the mass satisfies m ϕ (cid:38) H ∼ − eV, where H is the current value of the Hubble constant. Scalars satisfying this condition beginto coherently oscillate with an amplitude set by an initial misalignment in the field. Inturn, this results in homogeneous energy densities that redshift with the scale factor a ( t )as a ( t ) − , just like non-relativistic matter.For masses below m ϕ (cid:46) − eV, small-scale structure formation is suppressed onobservable length scales [45, 56, 57]. However, a scalar with a light mass can still makeup a fraction of the dark matter [55, 58]. In the range of masses we will be interested in,the scalar may contribute to dark energy for some period of time after matter-radiationequality, before contributing to dark matter. This can change the heights of the acousticpeaks, meaning that measurements of the cosmic microwave background (CMB) are alsostrongly constraining in this regime. For masses between 10 − eV (cid:46) m ϕ (cid:46) − eV, thescalar can only make up about 5% of the dark matter energy density at 95% C.L. [55].Taking these constraints into account, we present our results in Figure 7. We see fromthe left plot that for a shift δα ∼ . − .
08 as required to obtain the required baryonasymmetry (see Figure 3), if c g = 1, the scalar must have a mass 10 − . eV (cid:46) m ϕ (cid:46) − . eV for 0 . (cid:46) f DM (cid:46) .
03. The minimum fraction of dark matter the scalar couldmake up is bounded by the requirement m ϕ (cid:38) H ∼ − eV, and is about f DM (cid:38) − .– 20 –rom the right plot, we see that relaxing the assumption that c g = 1 but imposing that f DM ∼ . − eV (cid:46) m ϕ (cid:46) − eV. If ultimatelyit turned out that the E¨ot-Wash bound on c g is O (10) as might be expected based onthe discussion above, this mass range could be extended up to m ϕ (cid:46) − eV, since theallowed fraction of dark matter would increase as well. Thus we find that our mechanismof scanning the weak coupling constant to be viable for a wide range of ultra-light scalarmasses, and a variety of fractions of DM. It has been shown that gauge couplings can be altered in the early universe via scalar fieldsthat acquire a vev [11]. Following this approach, we interpret ϕ , in Equation (2.1) astwo real scalar fields, that are in a symmetric phase, with (cid:104) ϕ i (cid:105) = 0, at high temperatures.The fields go through a symmetry breaking transition at a temperature T sb (cid:38)
100 GeV,at which point they gain a non-zero vev (cid:104) ϕ i (cid:105) . (The two symmetry-breaking scales andthe vevs might be generated through separate mechanisms and does not necessarily havea common source.) Due to this contribution, the gauge couplings are different than theirSM values at the EW transition temperature.In the simplest scenario the dimension-5 terms in Equation (2.1) are generated byintegrating out N f vector-like fermions in the representation R of the gauge group, withmass M f and a Yukawa coupling y f , charged under either SU (2) L or SU (3) c . In this casewe have c g i (cid:104) ϕ i (cid:105) Λ i (cid:39) N f C ( R ) y f α i (cid:104) ϕ i (cid:105) M f . (5.20)The invariant C ( R ) is related to the quadratic Casimir operator, C ( R ), through the rela-tionship C ( R ) = d A d R C ( R ), where d A , d R are the dimensions of the adjoint and R respec-tively. For simplicity we turn on the interactions in Equation (2.1) one at a time, keepingone gauge coupling at its SM value while we change the other one. (See Figure 8.) Changing α : As discussed in Section 4, successful baryogenesis in the SMEFTscenario we consider can be achieved with a larger than expected α at T c (cid:39)
100 GeV.As can be seen in Figure 3, for δα = 0, successful baryogenesis requires δα (cid:38) .
01. Wecan see from Figure 8 that this requires (cid:12)(cid:12)(cid:12) c g (cid:104) ϕ (cid:105) Λ (cid:12)(cid:12)(cid:12) (cid:39) .
2. Taking y f (cid:39) , (cid:12)(cid:12)(cid:12) (cid:104) ϕ (cid:105) M f (cid:12)(cid:12)(cid:12) (cid:39)
1, thischange in α requires N f C ( R ) (cid:38)
270 fermions charged under SU (2) L with M f ∼ O (TeV).Although it has been argued that SU (2) L has an asymptotically safe fixed point withsuch a large number of weak-scale fermions, the UV behavior of this scenario is not wellunderstood. For example, it is expected that the theory be confined at very high scales,without a clear expectation to go into an unconfined phase below the Planck scale. Thus,within this scenario, we do not let the weak coupling to change from its SM value.We also note that α can be altered via the RG running, by adding fermions chargedsolely under SU (2) L . However, the required number of new charged fermions is vastly morethan allowed by constraints on the EW precision parameter W , which was constrained byLEP to be smaller than ∆ W (cid:46) × − . Additionally, strong LHC constraints can be– 21 – - Figure 8 : The absolute value of the dimension-5 operators in Equation (2.1) for a givenchange in α , . In this scenario we assume the scalar field ϕ , acquires a vev (cid:104) ϕ , (cid:105) beforethe EW phase transition, shifting the gauge couplings by δα , . In the purple shaded (onthe right half of the plot), the BAU is produced without any change in α , while in theblue shaded region (on the left half of the plot) BAU is produced without changing α .Note that the produced asymmetry could be larger than observed in these regions.placed on new SU (2) L charged fermions through modifications to the Drell-Yan process[59]. Changing α : On the other hand, the BAU can also be generated by decreasing α during the EW phase transition. In this case, with δα = 0, one needs δα (cid:39) − .
03, whichis achieved for (cid:12)(cid:12)(cid:12) c g (cid:104) ϕ (cid:105) Λ (cid:12)(cid:12)(cid:12) (cid:39) .
4. For y f (cid:39) , (cid:12)(cid:12)(cid:12) (cid:104) ϕ (cid:105) M f (cid:12)(cid:12)(cid:12) (cid:39)
1, and N f C ( R ) (cid:39)
60, much less thanthe number of fermions required to change α by an amount that would generate the BAU.Such new fermions would likely be vector-like, in which case strong limits of M f (cid:38) O (TeV)apply from searches at the LHC if they have either U (1) Y or SU (2) L charges (see e.g.[60–63]).The couplings must return to their SM values sometime before BBN since the formationof light nuclei is well described within the SM. This can be achieved by another transitionthat restores the symmetry that was broken by (cid:104) ϕ i (cid:105) at a temperature T sr < T sb . Weassume this symmetry restoration happens before BBN, T sr (cid:38) MeV. (We will be agnosticas to whether these transitions are first order or not.) Thus, although the gauge couplingsmight differ from their SM values for a period of time in the early universe, we recover theSM before BBN. Such a symmetry breaking pattern can be present if there are multiplescalar fields which go under symmetry-breaking and restoring phases. (Similar symmetrybreaking patterns have been studied in the literature, see e.g. [64, 65].) A more detailedwork on such a scenario is in progress [66].
We study the effects of varying weak and strong couplings in the early universe on theproduction of the observed baryon asymmetry. We do this in the context of the StandardModel with additional non-renormalisable operators, with the Lagrangian given in Equa-– 22 – φ ⟩ = 0⟨ φ ⟩ = 0⟨ φ ⟩ ≠ 0 Temperature δα ≠ 0 α = α SM Gauge couplings vary from their SM values due to contributions from fields that acquire vacuum expectation valuesThe symmetry is restored and the gauge couplings return to their SM values before BBN T sb T EW T sr T BBN
Figure 9 : An example of the cosmological history that is expected in the symmetry-breaking scenario.tion (2.1). We show that by raising the weak coupling constant and/or lowering the strongcoupling constant around the EW phase transition scale we can easily produce the requiredbaryon asymmetry. We do this in a model that was previously ruled out for baryogenesispurposes to demonstrate the power of these variations. The main reason for this success isthat a larger weak coupling constant raises the weak sphaleron rate, which in turn producesa larger baryon asymmetry, despite the risk of increased washout by the same sphalerons inthe broken phase of EW symmetry. Hence, the required extra CP violation can be smallercompared to more traditional models. Similarly, lowering the strong coupling constantweakens the strong sphalerons, preventing the washout of a chiral symmetry, which, again,raises the produced baryon asymmetry. These effects can be seen in Figures 2 to 4. Theviable parameter space can be almost entirely probed at HL-LHC, and entirely covered atproposed future lepton colliders.We identify two types of models that would generate deviations in weak and strongcoupling constants in the early universe. In one of these models, a light scalar couplesto the SU (2) L and U (1) Y field strengths. This scalar field could constitute part of thedark matter and the couplings are constrained by various astrophysical and fifth-forceexperiments. Another model relies on a scalar field that undergoes symmetry breakingand symmetry restoration phases in the early universe. This model requires O (50 − O (TeV) masses, that are charged under SU (3) c or SU (2) L and is expectedto have collider signatures. Acknowledgements
We thank Asher Berlin for helpful discussions. We thank David Morrissey for his thor-ough comments on a draft of this manuscript. SARE is supported in part by the U.S.Department of Energy under Contract No. DE-AC02-76SF00515, and in part by the Swiss– 23 –ational Science Foundation (SNF) project P2SKP2 171767. SI acknowledges supportfrom the University Office of the President via a UC Presidential Postdoctoral fellowshipand partial support from NSF Grant No. PHY-1620638. TRIUMF receives federal fund-ing via a contribution agreement with the National Research Council of Canada and theNatural Science and Engineering Research Council of Canada. This work was initiatedand partly performed at the Aspen Center for Physics, which is supported by NSF grantPHY-1607611. We thank the participants of the workshop “Understanding the Origin ofthe Baryon Asymmetry of the Universe” for excellent talks and discussions.While this work was in completion, [67] appeared on arXiv, which has overlap with Sec-tion 5.1.
A Gravitational Wave Signals with a Varying Weak Coupling Constant
It is well known that a strong first-order EW phase transition in the early universe producesgravitational waves (GWs) that are potentially visible at LISA [68, 69] and DECIGO [70].In this section we study the GW spectrum produced in an EW phase transition, which ismodified by both a | H | term and by a change in δα .The gravitational wave spectrum from a first-order cosmic phase transition includesthree contributions [71] Ω( f ) h = Ω col h + Ω sw h + Ω turb h . (A.1)Here the first term is generated via the collisions of bubbles. The soundwave contributionis due to the interactions of the bubbles with the plasma while the last term is due toturbulence. The sound wave contribution usually dominates when the Lorentz factor forthe advancing bubble wall does not diverge [72]. Indeed this is the regime we find ourselvesin our scenario [73–77]. The spectrum for the sound wave contribution is given by [72, 78] h Ω sw = 8 . × − (cid:18) g ∗ (cid:19) − / (cid:18) βH (cid:19) − Γ ¯ U f v w S sw ( f ) (A.2)where g ∗ is the number of degrees of freedom, Γ ∼ / β/H describes the inverse duration of the phase transition with respect to the Hubble rate H .The rms fluid velocity is given by ¯ U f ∼ (3 / κ f α , where α defines the strength of thephase transition as the ratio of the latent heat and the entropy. We estimate the bubblewall velocity v w ≈ . c , for which the efficiency is well approximated by [79] κ f ∼ α / .
017 + (0 .
997 + α ) / . (A.3)The spectral shape is given by S sw = (cid:18) ff sw (cid:19)
74 + 3 (cid:16) ff sw (cid:17) / (A.4)– 24 – - - - - - - Figure 10 : Gravitational wave power spectrum generated by Λ / √ c = 560 GeV (dark)and 600 GeV (light) and δα = 0 , . , . v w ∼ .
5. Note that δα = 0 . f sw = 8 . × − Hz (cid:18) v w (cid:19) (cid:18) βH (cid:19) (cid:18) T N GeV (cid:19) (cid:16) g ∗ (cid:17) / (cid:16) z p (cid:17) , (A.5)where T N is the nucleation temperature and z p is a simulation factor which we take tobe 6 . | H | operator, the sound waves do not last longer than the Hubble time [76, 77].This implies that simulations have overestimated the strength of the transition by a factorof v w (8 π ) / (cid:18) βH (cid:19) − ακ f ∼ O (10 ) . (A.6)We present the GW spectrum with this suppression factor in the sound wave peak ampli-tude in Figure 10 for various values of δα . Note that by including this suppression factor,we are presenting a pessimistic scenario. Recent simulations point to a more optimisticcase.The action for a critical bubble can be found using the public package BUBBLE PROFILER [80,81]. From the action as a function of temperature and δα , we calculate the nucleationtemperature and β/H . The general trend is that the peak amplitude reaches a maximalvalue for a moderate boost to δα , beyond which the amplitude is suppressed. We con-jecture the following explanation for this behavior. There is a competition between twoeffects when varying α : (i) the thermal barrier is larger due to the contribution from alarger coupling and (ii) the Higgs potential evolves faster due to larger contributions tothe thermal masses. The former effect enhances the GW signal while the latter suppressesit. This means there is an optimum value of α that maximizes the GW strength.– 25 – Details of the Baryon Asymmetry Production
The calculation of the final baryon asymmetry can only be performed rigorously in toymodels [30]. The approximate framework we use is known as the vev-insertion method.This method has a number of assumptions that have never been properly tested, althoughthere is ongoing work in this regard. These assumptions are as follows. • We can use the mass basis of the symmetric phase and ignore flavor oscillations. Thisis the assumption that most of the baryon asymmetry is produced in front of the wallwhere the vev is small and the mass matrix for quarks is basically diagonal with onlythe thermal masses. • The dominant source of CP violation comes from the collision term in the Boltzmannequations. We treat the semi-classical force as sub dominant. If the first assumptionis valid then this second assumption appears to be also valid since this source of CP violation tends to be couple orders of magnitude larger than the semi classical force.The baryon asymmetry is then calculated in three steps.1. Calculate the bubble wall dynamics.2. Use the output of step 1 to input into a set of transport dynamics which governsthe behavior of number densities near a bubble wall. This will result in calculatinga net chiral asymmetry that results from the CP -violating source. Such a chiralasymmetry will be relaxed by strong sphalerons which wash out chiral asymmetries.It also acts as a seed to bias the weak sphalerons. That is, weak sphalerons convertthe chiral asymmetry to a baryon asymmetry with an efficiency controlled by theweak sphaleron rate.3. Calculate EDM observables to make sure the CP -violating parameters are not ruledout by experiments. B.1 Step 1: Phase Transition
First one needs to calculate the effective potential at finite temperature. This is achieved bycalculating the one loop corrections to the effective potential using the finite temperaturepropagators. The effective potential can be written as a sum of zero temperature and finitetemperature pieces, V ( h, T ) = V ( h ) + V CW ( h ) + V T ( h, T ) (B.1)where the first term is the tree level potential and middle term is the zero temperature,Coleman-Weinberg loop correction. Loop corrections will include all particles that interactwith the Higgs. However, it is enough to focus on the ones with largest couplings, i.e. Higgs self-coupling, gauge bosons and the top quark. There is also a question of gauge-dependence, as the calculations are necessarily done in a certain gauge choice. Physicalquantities like the sphaleron energy is gauge-independent, while the measure of the order– 26 –f the phase transition, namely v c /T c is gauge-dependent. Although a gauge independentproxy has been developed [33], it is not necessary to worry about this numerically.The finite temperature corrections to the tree level potential are V T = (cid:88) i ∈ bosons n i T π J B (cid:20) m i + Π i T (cid:21) + (cid:88) i ∈ fermions n i T π J F (cid:20) m i T (cid:21) . (B.2)In the above the Π i terms are Debye masses that result from a resummation and areincluded to prevent the breakdown of perturbation theory. See, e.g. , [82]. Ignoring temper-ature corrections to the cosmological constant we can write a high temperature expansionfor the thermal functions that are correct up to m/T (cid:46) J F ( m /T ) ∼ − π m T , J B ∼ π m T − π m T . (B.3)We have dropped a log term which cancels the CW potential for a judicious value of therenormalization scale µ ∼ T .At very high temperatures, the Higgs is in a symmetric phase with v = 0. As thetemperature drops, the potential acquires a second minima at non-zero vev. At a criticaltemperature T c , these two minima become degenerate. The proxy for the strength of thephase transition is the ratio of the vev at T c to the critical temperature. The criticaltemperature is calculated by dVdh (cid:12)(cid:12)(cid:12)(cid:12) h = v c ,T = T c = 0 , V ( v c , T c ) = V (0 , T c ) . (B.4)In order to obtain a large BAU, one requires v c T c >
1. In the SM this is approximated as v c T c ∼ . g + δg ) g . (B.5)In principle this suggests that a large δg can catalyze a strongly first order phase transition.In practice, it is not possible to get larger than v c /T c ∼ / δg .Another way to get a strongly first order transition is by having a tree level barrierbetween the true and false vacuum that persists at zero temperature. This can be achievedby a single non-renormalizable operator added to the potential V = µ h − λ H h + c h . (B.6)The alternating signs between the different powers of the Higgs fields allows for there tobe a barrier between the true and false vacuum even at zero temperature.The last quantity we need to calculate is the bubble wall profile. This comes fromfinding the bounce solution to the classical equations of motion h (cid:48)(cid:48) + 2 r h (cid:48) = dVdh , (B.7)where the prime denotes a radial spatial derivative. The non-trivial solution to this equationis the bounce solution where one varies continuously from the true vacuum to the false one.– 27 –he spatial profile of this solution approximates a tanh solution with three parameters:a bubble wall width L w , the offset δ and the field value deep within the bubble h . Wefeed a numerical fit of the tanh profile into the transport equations described in the nextsection. This means the bubble wall parameters become functions of the temperature-varying couplings, i.e. L w → L w ( δα ), etc. The bubble wall profile needs to be calculatedat the nucleation temperature T N . This temperature is defined when the action evaluatedat the bounce satisfies S E T N ∼ − T N GeV − g ∗ , (B.8)where g ∗ is the number of relativistic degrees of freedom. This condition implies there isat least one critical bubble in the Hubble volume. B.2 Transport Equations
In calculating the production of the baryon asymmetry, we split the calculation into twosteps. First we calculate the baryon-conserving but CP -violating interactions with thebubble wall including baryon- and CP -conserving diffusion, scattering and decays. Thisallows one to calculate a profile for the total left-handed asymmetry. Then we feed thissolution into the transport equation with weak sphalerons. Separating the calculation intotwo steps is justified on the grounds of the typical timescales involved in the first step τ int ∼ /T is small compared to the time scale that governs baryon production t sph ∼ / Γ sph .Remarkably even when this is not true, the separation of the calculation into two stepsonly produces a few percent error.The transport equations that govern the first step given in the rest frame of the bubblewall for the right-handed top quark density n t , the left-handed top doublet number density n Q and the Higgs density n H are v w n (cid:48) t − D t n (cid:48)(cid:48) t = Γ m (cid:18) n Q k Q − n t k t (cid:19) − Γ Y (cid:18) n t k t − n H k H − n Q k Q (cid:19) (B.9)+ Γ SS (cid:18) n Q k Q − n t k t + 9( n Q + n t ) k B (cid:19) + S CP V ( z ) v w n (cid:48) Q − D Q n (cid:48)(cid:48) Q = − Γ m (cid:18) n Q k Q − n t k t (cid:19) + Γ Y (cid:18) n t k t − n H k H − n Q k Q (cid:19) (B.10) − ss (cid:18) n Q k Q − n t k t + 9( n Q + n t ) k B (cid:19) − S CP V ( z ) v w n (cid:48) H − D H n (cid:48)(cid:48) H = Γ Y (cid:18) n t k t − n H k H − n Q k Q (cid:19) (B.11)In the above we just guess the bubble wall velocity v w even though in reality it will dependon α and α . We set v w = 0 .
5. The k factors are factors that relate the number densitiesto their chemical potentials. They turn out to be not important. The Yukawa term Γ Y is usually dominated by the “scattering” contribution (decays tend to be kinematicallysuppressed) which involves a gluon and top annihilating into a Higgs boson and a topquark. This contribution is given byΓ Y ∼ × . g π T . (B.12)– 28 –he strong sphaleron rate that washes out the chiral asymmetry is simply given byΓ ss = 132 α T . (B.13)The diffusion coefficients for the top quark are very sensitive to α and α . They aregiven by [83] (the discussion before Equation (130) in particular). D − Q = (cid:15) L,R π α T log (cid:18) T M W (cid:19) + Y t π α tan θ w T log (cid:18) T M B (cid:19) + 807 π α T log (cid:18) T M G (cid:19) , (B.14)where M W = T (cid:112) πα / M G = T √ πα and M B = T (cid:112) πα tan θ /
3. Finally wehave the interactions with the space time varying Higgs. Specifically these interactions arethe left- and right-handed top quarks interacting with the vacuum. These are functionsof the bubble wall profile (described in the previous section), the thermal widths through E x = ω x − i Γ x where Γ x is the thermal width, and the thermal masses. The CP -violatinginteractions with the bubble wall are given by S CP Vt ( z ) = 3 v W y t v n ( z ) ∂ z v n ( z ) π (cid:90) k dkω L ω R Im (cid:20) ( E L E R + k ) n F ( E L ) + n F ( E R )( E L + E R ) + ( E L E ∗ R − k ) n F ( E L ) − n F ( E ∗ R )( E R − E L ) (cid:21) (B.15)where we have removed a divergent term by normal ordering [84]. Note that we neglecthole modes in the plasma. Due to this omission, we expect to underestimate the baryonasymmetry by a small amount [85, 86]. Similarly the relaxation term that describes CP -conserving interactions with the bubble wall is given byΓ m = 3 y t v n π T (cid:90) k dkω L ω R Im (cid:20) ( E L E R + k ) h F ( E L ) + h F ( E R )( E L + E R ) + ( E L E ∗ R − k ) h F ( E L ) + h F ( E ∗ R )( E R − E L ) (cid:21) . (B.16)The relevant thermal widths are Γ L,R = 43 α T . (B.17)and the thermal masses are given by m L = T (cid:114) g g + 1288 g + 116 y t + 116 y b (B.18) m R = T (cid:114) g g + 18 y t . (B.19)Both the CP -conserving and CP -violating interactions with the bubble wall have a resonantenhancement when m L ∼ m R . The width and height of the resonance are controlled bythe thermal width and the peak of the resonance is achieved when the thermal masses aredegenerate. – 29 –he second step in the calculation is in calculating the effect of weak sphalerons. Forthis we have a single transport equation D Q ρ (cid:48)(cid:48) B − v w ρ (cid:48) B Θ[ − z ]Γ WS = Θ[ − z ] 32 Γ WS (5 n Q ( z ) + 4 n t ( z )) (B.20)where Γ sph = 120 α T . The baryon yield is then calculated by solving the above equationand dividing by the entropy density Y B = 3Γ WS sD Q κ + (cid:90) −∞ dy n L ( y ) e − κ − y (B.21)with κ ± = ( v w ± (cid:112) v w + 15 D Q Γ W S ) / (2 D Q ). To a very good approximation the left-handednumber density in the symmetric phase can be approximated by a sum of exponentials, n L ( y ) ≈ (cid:80) i A i exp B i y [87]. The baryon asymmetry is then Y B = 3Γ WS sD Q ( v w + (cid:112) v w + 15 D Q Γ ws ) / (2 D Q ) (cid:88) i A i B i − ( v w − (cid:112) v w + 15 D Q Γ WS ) / (2 D Q ) . (B.22) References [1]
Planck collaboration, N. Aghanim et al.,
Planck 2018 results. VI. Cosmological parameters , .[2] S. Riemer-Sørensen and E. S. Jenssen, Nucleosynthesis Predictions and High-PrecisionDeuterium Measurements , Universe (2017) 44, [ ].[3] A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of theuniverse , Pisma Zh. Eksp. Teor. Fiz. (1967) 32–35.[4] D. E. Morrissey and M. J. Ramsey-Musolf, Electroweak baryogenesis , New J. Phys. (2012)125003, [ ].[5] G. A. White, A Pedagogical Introduction to Electroweak Baryogenesis . IOP Concise Physics.Morgan & Claypool, 2016, 10.1088/978-1-6817-4457-5.[6] M. Berkooz, Y. Nir and T. Volansky,
Baryogenesis from the Kobayashi-Maskawa phase , Phys. Rev. Lett. (2004) 051301, [ hep-ph/0401012 ].[7] I. Baldes, T. Konstandin and G. Servant, Flavor Cosmology: Dynamical Yukawas in theFroggatt-Nielsen Mechanism , JHEP (2016) 073, [ ].[8] I. Baldes, T. Konstandin and G. Servant, A first-order electroweak phase transition fromvarying Yukawas , Phys. Lett.
B786 (2018) 373–377, [ ].[9] B. von Harling and G. Servant,
Cosmological evolution of Yukawa couplings: the 5Dperspective , JHEP (2017) 077, [ ].[10] S. Bruggisser, T. Konstandin and G. Servant, CP-violation for Electroweak Baryogenesisfrom Dynamical CKM Matrix , JCAP (2017) 034, [ ].[11] S. Ipek and T. M. P. Tait,
An Early Cosmological Period of QCD Confinement , .[12] C. Grojean, G. Servant and J. D. Wells, First-order electroweak phase transition in thestandard model with a low cutoff , Phys. Rev.
D71 (2005) 036001, [ hep-ph/0407019 ]. – 30 –
13] C. Delaunay, C. Grojean and J. D. Wells,
Dynamics of Non-renormalizable ElectroweakSymmetry Breaking , JHEP (2008) 029, [ ].[14] J. de Vries, M. Postma, J. van de Vis and G. White, Electroweak Baryogenesis and theStandard Model Effective Field Theory , JHEP (2018) 089, [ ].[15] C. Balazs, G. White and J. Yue, Effective field theory, electric dipole moments andelectroweak baryogenesis , JHEP (2017) 030, [ ].[16] M. Chala, C. Krause and G. Nardini, Signals of the electroweak phase transition at collidersand gravitational wave observatories , JHEP (2018) 062, [ ].[17] J. De Vries, M. Postma and J. van de Vis, The role of leptons in electroweak baryogenesis , JHEP (2019) 024, [ ].[18] W. Altmannshofer, R. Harnik and J. Zupan, Low Energy Probes of PeV Scale Sfermions , JHEP (2013) 202, [ ].[19] G. Isidori, Flavor physics and CP violation , in
Proceedings, 2012 European School ofHigh-Energy Physics (ESHEP 2012): La Pommeraye, Anjou, France, June 06-19, 2012 ,pp. 69–105, 2014. . DOI.[20] P. Huet and E. Sather,
Electroweak baryogenesis and standard model CP violation , Phys.Rev.
D51 (1995) 379–394, [ hep-ph/9404302 ].[21] K. Kainulainen, V. Keus, L. Niemi, K. Rummukainen, T. V. I. Tenkanen and V. Vaskonen,
On the validity of perturbative studies of the electroweak phase transition in the Two HiggsDoublet model , .[22] R. R. Parwani, Resummation in a hot scalar field theory , Phys. Rev.
D45 (1992) 4695,[ hep-ph/9204216 ].[23] P. B. Arnold and O. Espinosa,
The Effective potential and first order phase transitions:Beyond leading-order , Phys. Rev.
D47 (1993) 3546, [ hep-ph/9212235 ].[24] D. Curtin, P. Meade and H. Ramani,
Thermal Resummation and Phase Transitions , Eur.Phys. J.
C78 (2018) 787, [ ].[25] S. Di Vita, G. Durieux, C. Grojean, J. Gu, Z. Liu, G. Panico et al.,
A global view on theHiggs self-coupling at lepton colliders , JHEP (2018) 178, [ ].[26] ACME collaboration, V. Andreev et al.,
Improved limit on the electric dipole moment of theelectron , Nature (2018) 355–360.[27] D. Bodeker, G. D. Moore and K. Rummukainen,
Chern-Simons number diffusion and hardthermal loops on the lattice , Phys. Rev.
D61 (2000) 056003, [ hep-ph/9907545 ].[28] G. D. Moore and M. Tassler,
The Sphaleron Rate in SU(N) Gauge Theory , JHEP (2011)105, [ ].[29] C. Lee, V. Cirigliano and M. J. Ramsey-Musolf, Resonant relaxation in electroweakbaryogenesis , Phys. Rev.
D71 (2005) 075010, [ hep-ph/0412354 ].[30] V. Cirigliano, C. Lee and S. Tulin,
Resonant Flavor Oscillations in ElectroweakBaryogenesis , Phys. Rev.
D84 (2011) 056006, [ ].[31] M. Carena, J. M. Moreno, M. Quiros, M. Seco and C. E. M. Wagner,
Supersymmetric CPviolating currents and electroweak baryogenesis , Nucl. Phys.
B599 (2001) 158–184,[ hep-ph/0011055 ]. – 31 –
32] J. M. Cline, M. Joyce and K. Kainulainen,
Supersymmetric electroweak baryogenesis , JHEP (2000) 018, [ hep-ph/0006119 ].[33] H. H. Patel and M. J. Ramsey-Musolf, Baryon Washout, Electroweak Phase Transition, andPerturbation Theory , JHEP (2011) 029, [ ].[34] L. Carson and L. D. McLerran, Approximate Computation of the Small FluctuationDeterminant Around a Sphaleron , Phys. Rev.
D41 (1990) 647.[35] L. Carson, X. Li, L. D. McLerran and R.-T. Wang,
Exact Computation of the SmallFluctuation Determinant Around a Sphaleron , Phys. Rev.
D42 (1990) 2127–2143.[36] F. R. Klinkhamer and N. S. Manton,
A Saddle Point Solution in the Weinberg-SalamTheory , Phys. Rev.
D30 (1984) 2212.[37] X. Gan, A. J. Long and L.-T. Wang,
Electroweak sphaleron with dimension-six operators , Phys. Rev.
D96 (2017) 115018, [ ].[38] J. Lim, J. R. Almond, M. A. Trigatzis, J. A. Devlin, N. J. Fitch, B. E. Sauer et al.,
LaserCooled YbF Molecules for Measuring the Electron’s Electric Dipole Moment , Physical ReviewLetters (Mar., 2018) 123201, [ ].[39] A. C. Vutha, M. Horbatsch and E. A. Hessels,
Oriented polar molecules in a solid inert-gasmatrix: a proposed method for measuring the electric dipole moment of the electron , .[40] I. Kozyryev and N. R. Hutzler, Precision Measurement of Time-Reversal Symmetry Violationwith Laser-Cooled Polyatomic Molecules , Phys. Rev. Lett. (2017) 133002, [ ].[41] M. S. Turner,
Coherent Scalar Field Oscillations in an Expanding Universe , Phys. Rev.
D28 (1983) 1243.[42] D. B. Kaplan and M. B. Wise,
Couplings of a light dilaton and violations of the equivalenceprinciple , JHEP (2000) 037, [ hep-ph/0008116 ].[43] T. Damour and J. F. Donoghue, Equivalence Principle Violations and Couplings of a LightDilaton , Phys. Rev.
D82 (2010) 084033, [ ].[44] T. Damour and J. F. Donoghue,
Phenomenology of the Equivalence Principle with LightScalars , Class. Quant. Grav. (2010) 202001, [ ].[45] J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Cosmology with ultralight pseudoNambu-Goldstone bosons , Phys. Rev. Lett. (1995) 2077–2080, [ astro-ph/9505060 ].[46] B. Bertotti, L. Iess and P. Tortora, A test of general relativity using radio links with theCassini spacecraft , Nature (2003) 374–376.[47] C. J. A. P. Martins,
The status of varying constants: a review of the physics, searches andimplications , .[48] J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, R. F. Carswell and M. B.Bainbridge, Indications of a spatial variation of the fine structure constant , Phys. Rev. Lett. (2011) 191101, [ ].[49] T. Damour and F. Dyson,
The Oklo bound on the time variation of the fine structureconstant revisited , Nucl. Phys.
B480 (1996) 37–54, [ hep-ph/9606486 ].[50] A. Coc, N. J. Nunes, K. A. Olive, J.-P. Uzan and E. Vangioni,
Coupled Variations ofFundamental Couplings and Primordial Nucleosynthesis , Phys. Rev.
D76 (2007) 023511,[ astro-ph/0610733 ]. – 32 –
51] S. Schlamminger, K. Y. Choi, T. A. Wagner, J. H. Gundlach and E. G. Adelberger,
Test ofthe equivalence principle using a rotating torsion balance , Phys. Rev. Lett. (2008)041101, [ ].[52] J. Khoury and A. Weltman,
Chameleon fields: Awaiting surprises for tests of gravity inspace , Phys. Rev. Lett. (2004) 171104, [ astro-ph/0309300 ].[53] N. Blinov, S. A. R. Ellis and A. Hook, Consequences of Fine-Tuning for Fifth ForceSearches , JHEP (2018) 029, [ ].[54] A. Hook and G. Marques-Tavares, Relaxation from particle production , JHEP (2016) 101,[ ].[55] R. Hlozek, D. Grin, D. J. E. Marsh and P. G. Ferreira, A search for ultralight axions usingprecision cosmological data , Phys. Rev.
D91 (2015) 103512, [ ].[56] K. Coble, S. Dodelson and J. A. Frieman,
Dynamical Lambda models of structure formation , Phys. Rev.
D55 (1997) 1851–1859, [ astro-ph/9608122 ].[57] W. Hu, R. Barkana and A. Gruzinov,
Cold and fuzzy dark matter , Phys. Rev. Lett. (2000)1158–1161, [ astro-ph/0003365 ].[58] L. Amendola and R. Barbieri, Dark matter from an ultra-light pseudo-Goldsone-boson , Phys.Lett.
B642 (2006) 192–196, [ hep-ph/0509257 ].[59] D. S. M. Alves, J. Galloway, J. T. Ruderman and J. R. Walsh,
Running ElectroweakCouplings as a Probe of New Physics , JHEP (2015) 007, [ ].[60] ATLAS collaboration, M. Aaboud et al.,
Combination of the searches for pair-producedvector-like partners of the third-generation quarks at √ s =
13 TeV with the ATLAS detector , Phys. Rev. Lett. (2018) 211801, [ ].[61]
ATLAS collaboration, M. Aaboud et al.,
Search for single production of vector-like quarksdecaying into
W b in pp collisions at √ s = 13 TeV with the ATLAS detector , Submitted to:JHEP (2018) , [ ].[62]
CMS collaboration, A. M. Sirunyan et al.,
Search for single production of vector-like quarksdecaying to a top quark and a W boson in proton-proton collisions at √ s =
13 TeV , Eur.Phys. J.
C79 (2019) 90, [ ].[63]
CMS collaboration, A. M. Sirunyan et al.,
Search for vector-like quarks in events with twooppositely charged leptons and jets in proton-proton collisions at √ s =
13 TeV , Eur. Phys. J.
C79 (2019) 364, [ ].[64] M. J. Baker and J. Kopp,
Dark Matter Decay between Phase Transitions at the Weak Scale , Phys. Rev. Lett. (2017) 061801, [ ].[65] M. J. Baker, M. Breitbach, J. Kopp and L. Mittnacht,
Dynamic Freeze-In: Impact ofThermal Masses and Cosmological Phase Transitions on Dark Matter Production , JHEP (2018) 114, [ ].[66] S. Ipek and T. M. Tait, Multiple Phase Transitions And Early QCD Confinement , work inprogress .[67] U. Danielsson, R. Enberg, G. Ingelman and T. Mandal, Varying gauge couplings and colliderphenomenology , .[68] A. Mazumdar and G. White, Cosmic phase transitions: their applications and experimentalsignatures , . – 33 –
69] C. Caprini and D. G. Figueroa,
Cosmological Backgrounds of Gravitational Waves , Class.Quant. Grav. (2018) 163001, [ ].[70] S. Kawamura et al., The Japanese space gravitational wave antenna: DECIGO , Class.Quant. Grav. (2011) 094011.[71] D. J. Weir, Gravitational waves from a first order electroweak phase transition: a briefreview , Phil. Trans. Roy. Soc. Lond.
A376 (2018) 20170126, [ ].[72] M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir,
Shape of the acousticgravitational wave power spectrum from a first order phase transition , Phys. Rev.
D96 (2017)103520, [ ].[73] D. Bodeker and G. D. Moore,
Can electroweak bubble walls run away? , JCAP (2009)009, [ ].[74] D. Bodeker and G. D. Moore,
Electroweak Bubble Wall Speed Limit , JCAP (2017) 025,[ ].[75] D. Croon, V. Sanz and G. White,
Model Discrimination in Gravitational Wave spectra fromDark Phase Transitions , JHEP (2018) 203, [ ].[76] J. Ellis, M. Lewicki and J. M. No, On the Maximal Strength of a First-Order ElectroweakPhase Transition and its Gravitational Wave Signal , .[77] J. Ellis, M. Lewicki, J. M. No and V. Vaskonen, Gravitational wave energy budget in stronglysupercooled phase transitions , .[78] M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Gravitational waves from thesound of a first order phase transition , Phys. Rev. Lett. (2014) 041301, [ ].[79] J. R. Espinosa, T. Konstandin, J. M. No and G. Servant,
Energy Budget of CosmologicalFirst-order Phase Transitions , JCAP (2010) 028, [ ].[80] S. Akula, C. Bal´azs and G. A. White,
Semi-analytic techniques for calculating bubble wallprofiles , Eur. Phys. J.
C76 (2016) 681, [ ].[81] P. Athron, C. Bal´azs, M. Bardsley, A. Fowlie, D. Harries and G. White,
BubbleProfiler:finding the field profile and action for cosmological phase transitions , .[82] L. Delle Rose, C. Marzo and A. Urbano, On the fate of the Standard Model at finitetemperature , JHEP (2016) 050, [ ].[83] M. Joyce, T. Prokopec and N. Turok, Nonlocal electroweak baryogenesis. Part 1: Thin wallregime , Phys. Rev.
D53 (1996) 2930–2957, [ hep-ph/9410281 ].[84] T. Liu, M. J. Ramsey-Musolf and J. Shu,
Electroweak Beautygenesis: From b → s CP-violationto the Cosmic Baryon Asymmetry , Phys. Rev. Lett. (2012) 221301, [ ].[85] H. A. Weldon,
Structure of the quark propagator at high temperature , Phys. Rev.
D61 (2000)036003, [ hep-ph/9908204 ].[86] S. Tulin and P. Winslow,
Anomalous B meson mixing and baryogenesis , Phys. Rev.
D84 (2011) 034013, [ ].[87] G. A. White,
General analytic methods for solving coupled transport equations: Fromcosmology to beyond , Phys. Rev.
D93 (2016) 043504, [ ].].