Wall speed and shape in singlet-assisted strong electroweak phase transitions
Avi Friedlander, Ian Banta, James M. Cline, David Tucker-Smith
WWall speed and shape in singlet-assistedstrong electroweak phase transitions
Avi Friedlander, ∗ Ian Banta, † James M. Cline, ‡ and David Tucker-Smith § Queen’s University, Department of Physics & Engineering PhysicsAstronomy Kingston, Ontario, K7L 3N6 Kingston, Canada Department of Physics, Williams College, Williamstown, MA 01267 McGill University, Department of Physics, 3600 University St., Montr´eal, QC H3A2T8 Canada (Dated: August 2020)Models with singlet fields coupling to the Higgs can enable a strongly first order electroweakphase transition, of interest for baryogenesis and gravity waves. We improve on previous attemptsto self-consistently solve for the bubble wall properties—wall speed v w and shape—in a highlypredictive class of models with Z -symmetric singlet potentials. A new algorithm is implemented todetermine v w and the wall profiles throughout the singlet parameter space in the case of subsonicwalls, focusing on models with strong enough phase transitions to satisfy the sphaleron washoutconstraint for electroweak baryogenesis. We find speeds as low as v w ∼ = 0 .
22 in our scan overparameter space, and the singlet must be relatively light to have a subsonic wall, m s (cid:46)
110 GeV.
I. INTRODUCTION
The electroweak phase transition (EWPT) in theearly universe has been intensively studied as a pos-sible source for the cosmic baryon symmetry andgravitational waves. Within the standard model(SM) neither of these interesting outcomes are pos-sible, given the known mass of the Higgs boson,because the phase transition is a smooth crossover[1, 2], whereas a first-order EWPT is required forelectroweak baryogenesis (EWBG) and productionof observable gravity waves (for reviews, see for ex-ample refs. [3, 4]). New physics, typically in theform of scalar fields coupling to the Higgs boson, canhowever lead to a first-order transition, with conse-quent nucleation of bubbles of the true (electroweaksymmetry broken) vacuum, at the onset of the tran-sition.In order to make quantitative predictions for ei-ther baryogenesis or gravitational wave productionin a given model, it is necessary to understand thedetailed properties of the phase transition bubbles,especially the shape of the bubble walls (typicallymodeled as a tanh with some thickness L w ) andthe terminal velocity v w attained by them, once theforces of internal pressure and external friction fromthe plasma have balanced each other. This calcula-tion, first carried out for the SM in refs. [5–7] (as-suming a light Higgs boson), and in the MinimalSupersymmetric Standard Model, where the phasetransition is enhanced by light stops, in refs. [8, 9], ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] turns out to be quite challenging because the fric-tional force, which requires solving the Boltzmannequations for the perturbations of the plasma causedby the wall, depends on the same wall propertiesthat one is trying to determine.A self-consistent procedure to solve this systemis numerically expensive, and for this reason manystudies of EWBG or gravitational wave productionleave L w and v w as phenomenological parametersthat can be freely varied, or in a somewhat betterapproximation, calculated by modeling the frictionin a phenomenological way [10–12]. However to as-sess the prospects for a specfic model to yield in-teresting results, one must eventually carry out theactual computation of L w and v w with the actualfriction term derived from the fluid perturbations.Accurate estimates of these parameters are neededto make quantitative predictions for baryogenesis orgravity wave production.The procedure becomes even more laborious inthe case where an extra singlet field couples to theHiggs, in order to facilitate the first order transition,and also gets a vacuum expectation value (VEV) inthe bubble wall [13, 14]. In that case one must solvefor both field profiles, which has been attempted inrefs. [15, 16], subject to some limiting approxima-tions. In particular, these previous works assumedthat the bubble wall shapes are described by tanhprofiles. In reality, the Higgs field and the singletcan have shapes that differ from such an assump-tion, and it is not obvious how strongly this affectsthe determination of v w . One of our main purposesis to overcome this limitation by developing an al-gorithm to determine the actual wall profiles alongwith v w .Moreover previous studies of singlet-assistedstrong EWPTs have focused on a few benchmarkmodels. In the present work we make a comprehen- a r X i v : . [ h e p - ph ] S e p sive scan of the parameter space for a class of mod-els, where the singlet potential has the Z symmetry s → − s , and the singlet VEV disappears at low tem-perataures. This choice has the virtue of simplicity,being characterized by three parameters, the singletmass m s , its cross coupling λ hs to the Higgs, and theVEV w of s in the false vacuum where h = 0. Thebarrier between the true and false vacua providedby the λ hs h s interaction is already present at treelevel, and is what enables the phase transition to bestrongly first order [13, 14]. Moreover with (cid:104) s (cid:105) = 0at T = 0, the new sources of CP violation needed forEWBG are not overly constrained by experimentallimits on electric dipole moments.The paper is organized as follows. Section II de-scribes the singlet scalar model used throughout thepaper. Section III outlines the main features of theelectroweak phase transition dynamics that will bestudied in detail in the following. In section IVthe methodology for determining the wall dynam-ics, including its velocity, are described; the resultsof those calculations are presented in section V. Con-clusions are given in section VI. Appendices con-tain details concerning the finite-temperature effec-tive potential (appendix A), diffusion equations usedto determine the fluid perturbations (appendix B),and exceptional models that have peculiar featuresin their potentials (appendix C). II. THE MODEL
A simple extension of the SM is the addition ofa scalar singlet s that couples only to the Higgsfield, and has the Z symmetry s → − s . Its zero-temperature tree level potential is given by V = λ h (cid:18) | H | − v (cid:19) + 14 λ s (cid:0) s − w (cid:1) + 12 λ hs | H | s (1)where H is the Standard Model Higgs doublet, and λ h , v are the Higgs self-coupling and VEV respec-tively. There are three new parameters λ s , w , and λ hs , that describe the singlet’s self coupling, its VEVwhen in the false minimum where H = 0, and thecoupling between H and s . There is no loss of gen-erality by omitting a separate m s mass term. Thephysical singlet mass in the electroweak broken vac-uum is given by m s = − λ s w + λ hs v (2)We restrict the parameters so that m s >
0, implyingthat (cid:104) s (cid:105) = 0 in the true vacuum. The Higgs doubletcomponents are H = ( χ + iχ , h + iχ ) T / √ h denotes the background Higgs field, and the χ ’s are the Goldstone bosons.The full effective potential takes into account one-loop corrections and temperature effects, V eff = V + V + V CT + V T (4)where V is the tree-level potential (1), V is theone-loop correction, V CT contains the countertermsassociated with V , and V T is the thermal contribu-tion, including ring resummation of thermal masses.These expressions are standard, and we have de-scribed them in detail in appendix A. V eff is deter-mined by the measured SM parameters and the threenew ones, that we henceforth take to be w , λ hs andthe singlet mass m s by trading λ s for m s througheq. (2). III. PHASE TRANSITION
Because of our assumption that (cid:104) s (cid:105) = 0 at lowtemperatures, the λ hs h s coupling creates a bar-rier in field space between the false and true vacua,and gives rise to a two-step phase transition. Attemperatures T (cid:29) w , the minimum of the potentialis at the origin, where both electroweak symmetryand the Z symmetry are restored. At some temper-ature T (cid:48) , the first transition occurs, where (cid:104) s (cid:105) → w (cid:48) (see fig. 1). As T decreases, (cid:104) s (cid:105) and the correspond-ing minimum of the potential becomes metastable.At the critical temperature T c , the two minima be-come degenerate, and at a slightly lower temperature T n nucleation of bubbles begins, signaling the sec-ond transition where electroweak symmetry is bro-ken while the Z symmetry is restored. These twotransitions are summarized as1. At T = T (cid:48) ( h, s ) : (0 , → (0 , w (cid:48) )2. At T = T n ( h, s ) : (0 , w n ) → ( v n ,
0) .It is the second transition that is important forbaryogenesis. We note that while domain walls could ′ T ′ w >T > w n T~100 GeV hstep 2(EWSB)step 1 w n n v s FIG. 1. Sequence of phase transitions in field space. form during the first transition, the restoration ofthe Z symmetry during the second transition willcause them to annihilate. This occurs long beforethey can dominate the energy density of the uni-verse, hence we expect that no cosmological prob-lems will arise from this brief appearance of domainwalls [13].The dynamics of the phase transition dependstrongly on T n , the temperature at which the prob-ability of a bubble nucleating within one Hubblevolume per Hubble time is O (1) [17]. The tunnel-ing probability goes as exp( − S /T ), where S is thethree-dimensional Euclidean action. We used Cos-moTransitions [18] to find phase transition candi-dates and to determine T n . The criterion for thenucleation temperature is taken to be S /T n = 140[17].Since our investigation is motivated by elec-troweak baryogenesis, we focus attention on first or-der transitions that are strong enough to preservethe baryon asymmetry from washout by residualsphaleron interaction inside the bubbles. This re-quires the Higgs VEV at the nucleation temperatureto satisfy [19] v n T n (cid:38) . . (5) IV. WALL DYNAMICS
The bubble-wall dynamics are determined by theinteractions of the Higgs and singlet fields with athermal fluid consisting of top quarks, electroweakgauge bosons, and any other particles to which thescalars couple significantly. After the bubble nucle-ates, it expands due to the outward pressure causedby the potential difference between the phases of thescalar fields on either side. The interactions of thewall with the surrounding fluid counteract the ex-pansion by a friction force that depends on the speedand shape of the wall. CosmoTransitions occasionally fails to find transitionswhen they should exist; in such cases, changing the value of λ hs by O (10 − ) can overcome the problem. Moreover, Cos-moTransitions often reports more phase transitions thanexpected for a given model; we find that defining the EWPTas the most recent first order transition where the Higgs’VEV in the unbroken phase is smaller than the nucleationtemperature gives correct results. Of those, only phasetransitions that ended with no singlet VEV were studied.We found it a useful cross-check to require the transitionidentified by CosmoTransitions to have a critical tempera-ture that matched our own calculations for the parameterpoint in question. If the friction is strong enough, the bubbles reacha steady-state velocity whose value is relevant forgravitational waves and baryogenesis. The termi-nal v w depends upon the field profiles that solve theequations of motion. These in turn depend upon thetemperature T w of the wall and the v w -dependentfriction exerted by the plasma on the wall, lead-ing to T w > T n , due to heating by the fluid. Aself-consistent solution thus requires simultaneouslysolving for v w and the scalar field profiles in the wall. IV.1. Deflagration profiles
The wall temperature, and the rest of the dynam-ics of the bubble, depend on whether the phase tran-sition proceeds through deflagrations or detonations(hybrids of these two are also possible). For subsonicbubble walls, with v w < / √
3, the bubbles grow viadeflagrations [10], in which the wall is preceded bya shock front that moves through the fluid, perturb-ing it, increasing the temperature from T n to T s andcausing the wall to move (see fig. 2). The fluid ve-locity decreases until the point where the wall passesit, so that the fluid behind the wall is at rest relativeto that preceding the shock front. In this work welimit our investigation to the case of deflagrations,hence subsonic walls, deferring the study of super-sonic walls to the future [20].Because of the heating, the bubble wall dynamicsare not determined at temperature T n , but ratherthe temperature of the fluid near the wall. The cal-culation is performed at times sufficiently long af-ter nucleation that the bubble has reached a steady-state velocity, and the profiles of the fluid pertur-bations vary on scales much larger than the wallthickness. Therefore one approximates the wall as adiscontinuity, such that the fluid temperature is T + ( T − ) just in front of (behind) the wall. The fluidvelocity likewise is discontinuous there.Since it is often convenient to switch between ref-erence frames, we adopt the notation v xy for thevelocity of x in the reference frame y . In this con-text, x and y either refer to the wall ( w ), shock front( s ), or the fluid at position 1 (in front of the wall),2 (behind the shock front), or u (the unperturbed“universe” frame, in front of the shock front or be-hind the wall). The wall velocity, which is measuredwith respect to the fluid directly in front of the wall,is v w in this notation. We note that for any x and y , v xy = − v yx . A diagram depicting the geometryand labels is shown in fig. 2.The relationships between the various fluid veloc-ities and temperatures are found by integrating thestress tensor T µν across either of the two interfacesshown in fig. 2. Approximating the fluid as perfect, Shock Front Bubble Wall h = v n S = 0h = 0S = w n T - T + T s T n Scalar VEVReference Frame LabelsTemperatures u 12 uws
FIG. 2. Illustration of the geometry of a deflagration. The bubble wall and shock front are moving to the left withthe inside of the bubble being on the right of the figure. these depend only on the fluid density and pressure.The equations of state can be expressed as [16] p ± = 13 a ± ( T n ) T ± − (cid:15) ± ( T n ) (6) ρ ± = a ± ( T n ) T ± + (cid:15) ± ( T n ) (7)where ρ ± is the fluid density on either side of thebubble wall and p ± is the pressure. a ± and (cid:15) ± aregiven by a ± ( T ) = − T d F ± ( T ) dT (8) (cid:15) ± ( T ) = F ± ( T ) + 13 a ± ( T ) T . (9) F ± is the free energy of the fluid evaluated at therespective VEVs outside and inside wall and T = T n .It is given by F ( h, s, T ) = V eff ( h, s, T ) − g (cid:48)∗ π T . (10)Here g (cid:48)∗ = 107 . − . .
25 is the effectivenumber of degrees of freedom, apart from the t , W/Z , h , χ and s , whose contributions are alreadyincluded in V eff . In general, a ± ( T ) and (cid:15) ± ( T ) in(6,7) should be evaluated at the temperatures T ± ,but for transitions typically of interest for baryogen-esis, where there is a limited degree of supercooling,the T -dependence of a ( T ) and (cid:15) ( T ) is insignificant. Integrating T µν across the bubble wall providesthe relations [10] v w v wu = p + − p − ρ + − ρ − , v w v wu = p + + ρ − ρ + + p − . (11)Integrating across the shock front, the temperaturechanges, but not the the field values, leading to v su v s = 13 , v su v s = T n + 3 T s T n + T s . (12)Fluid velocities in the wall and shock wave frameare related by Lorentz transforming to the u frameusing v u = v su − v s − v su v s , v u = v wu − v w − v wu v w . (13)The relationship between the fluid velocity andtemperature behind the shock front and in frontof the wall can be approximated by linearizing thestress-energy equations with respect to small fluidvelocities in the universe frame [11], to obtain T s = T + exp (cid:20) − v u v wu − v wu (cid:18) v wu − v su (cid:19)(cid:21) (14) v u = v u (cid:18) v wu v su (cid:19) (cid:18) v su − v wu − (cid:19) (15)For a given guess of the wall velocity, v w ≡ v w , andan associated nucleation temperature, eqs. (11-15)can be used to solve for the eight remaining vari-ables: T + , T − , T s , v wu , v su , v u , v u , and v s . Anexample of the solution to these equations for sampleset of parameters is shown in fig. 3. v w T ( G e V ) T + TT n T c FIG. 3. An example of how the wall temperaturechanges as a function of v w for a sample model with m s = 63 GeV, w = 130 GeV, and λ hs = 0 . IV.2. Equations of Motion
The equation of motion of a scalar field coupledto a perfect fluid has been derived by enforcing theconservation of the stress-energy tensor in the WKBapproximation [7], or starting from the Kadanoff-Baym equations [15]. Both methods lead to (cid:3) φ + ∂V ( φ ) dφ + (cid:88) i n i dm i dφ (cid:90) d p (2 π ) E f i ( (cid:126)p, x ) = 0(16)where V ( φ ) is the zero-temperature effective poten-tial, the sum is over all particles that couple to φ , n i is the number of degrees of freedom of particle i , m i is its field-dependent mass, and f i ( (cid:126)p, x ) is its phasespace distribution. By separating f i = f ,i + δf i intoequilibrium and out-of-equilibrium components, theequation of motion takes a more useful form. Theintegral over f ,i is equivalent to accounting for the T -dependence of the effective potential, giving (cid:3) φ + ∂V eff ( φ, T ) dφ + (cid:88) i n i dm i dφ (cid:90) d p (2 π ) E δf i ( (cid:126)p, x ) = 0(17)The third term in (17) describes the friction forcethat comes from the dissipative interactions betweenthe scalar field and the surrounding fluid.In the following, we assume that the dominantsources of friction are the top quark and electroweakgauge bosons. The lighter fermions, gluons and pho-tons can be safely ignored because of their negligiblecouplings to the Higgs field. The self-couplings andmixing of the scalar fields are assumed to be sub-dominant due to their fewer degrees of freedom rel-ative to vector bosons or quarks. It is possible thatincluding these contributions would lead to moder- ately slower walls, which could be advantageous forbaryogenesis, but we defer this issue to future study.For the electroweak phase transition consideredhere, there are two relevant scalar fields, each withits own equation of motion, that must be simultane-ously solved. Eq. (17) can be further simplified byaccounting for the spherical symmetry of the walland going to the planar limit, which reduces thesystem to one spatial dimension, and by consideringonly the steady-state regime. Therefore, the equa-tions of motion for a bubble wall traveling in thenegative z direction become − h (cid:48)(cid:48) ( z ) + ∂V eff ( h, s, T ) ∂h (18)+ (cid:88) i = t,W,Z n i dm i dh (cid:90) d p (2 π ) E δf i ( (cid:126)p, z ) = 0 − s (cid:48)(cid:48) ( z ) + ∂V eff ( h, s, T ) ∂s = 0 (19)where primes denote derivatives with respect to z .Strictly speaking, the existence of a steady-state so-lution to the equations of motion does not guaranteethat those solutions will in fact be realized in thephysical setting, but this issue is beyond the scopeof this paper. IV.3. Friction
The friction experienced by the wall depends on δf i ( (cid:126)p, z ), the deviation from equilibrium of W/Z and t . We adopt the fluid approximation framework de-veloped in [7], in which the friction is fully describedby three fluids: that of the top quark, the massivegauge bosons, and the other particles, denoted as the“background.” We label the gauge boson contribu-tion by W although it also includes Z . For simplicity W and Z are grouped together due to their similarcouplings, and assigned a mass-squared that is theweighted average of m W and m Z . The backgroundfluid encompasses all the fields that are assumedto contribute negligible friction, but which never-theless play an important role in the wall dynam-ics. We consider friction only from fluid excitationswith large momentum, such that the wavelength isshorter than the width of the wall. It has been shownthat IR excitations in the massive gauge boson fluidcan be important [21], but we have checked numer-ically that these are subdominant for parameters ofinterest in the present study, using the same approx-imations to evaluate the IR contributions as in ref.[16], but taking care to impose the perturbative cut-off m W ( z ) > g T [21].The phase space distribution for the t and W flu-ids can be parametrized as f i ( E, z ) = 1 e ( E + δ i ( z )) /T ± / − is for fermions/bosons and δ i ( z ) = − (cid:104) T ( δµ i + δµ bg )( z ) + E ( δτ i + δτ bg )( z )+ p z ( δv i + δv bg )( z ) (cid:105) (21)accounts for perturbations in the fluids. δµ i ( z ), δτ i ( z ), δv i ( z ) are respectively the perturbations inthe chemical potential, the relative temperature andthe velocity. The subscript bg denotes the back-ground fluid. All the perturbations are relative tothe fluid directly in front of the wall where µ = 0and T = T + as described in section IV.1.Deviations from equilibrium in the fluids are gov-erned by the Boltzmann equation ddt f i ( E, z ) = − C [ f i ( E, z )] . (22)Rather than solving the full Boltzmann equation,one linearizes it in δ i ( z ), and converts it to a sys-tem of ordinary differential equations by takingthree moments: (cid:82) d p/ (2 π ) , (cid:82) ( E/T ) d p/ (2 π ) ,and (cid:82) p z d p/ (2 π ) . A detailed derivation is pro-vided in appendix B, leading to the coupled matrixequations A W ( (cid:126)q W + (cid:126)q bg ) (cid:48) + Γ W (cid:126)q W = S W (23) A t ( (cid:126)q t + (cid:126)q bg ) (cid:48) + Γ t (cid:126)q t = S t (24) A bg (cid:126)q (cid:48) bg + Γ bg, W (cid:126)q W + Γ bg,t (cid:126)q t = 0 (25)where (cid:126)q Ti = ( δµ i , δτ i , δv i ). The A i matrices for i = W, t take the form A i ≡ v w c i v w c i d i v w c i v w c i d i d i d i v w d i (26)while the source terms are S i ≡ m (cid:48) i m i T v w c i v w c i (27)The coefficients c ij and d ij denote the integrals c ij (cid:16) m i T (cid:17) ≡ (cid:90) d p (2 π ) (cid:0) − f (cid:48) ,i (cid:1) E j − T j +1 (28)and d ij (cid:16) m i T (cid:17) ≡ (cid:90) d p (2 π ) (cid:0) − f (cid:48) ,i (cid:1) p E j − T j +1 (29) where f ,i is the equilibrium distribution functionfor particle i . In previous literature (with the excep-tion of ref. [22]), these coefficients were evaluated inthe massless approximation, where d ij = c ij , setting m i /T = 0. But for some phase transitions satis-fying the sphaleron bound (5), m t /T > A bg = 20 A W | m =0 + 78 A t | m =0 (30)where the background fluids are approximated asmassless [15].The Γ i matrices in eqs. (23, 24) quantify the in-teractions of the fluids. Γ t and Γ W take the formΓ i ≡ T Γ µ i Γ δT i µ i Γ δT i
00 0 Γ vi (31)where the matrix elements are numerical constantscalculated in [7] to be Γ W /T = . . . . . Γ t /T = . . . . . (32)Using energy-momentum conservation, the back-ground fluid collision terms are given byΓ bg,i = − n i Γ i (33)where n t = 12 and n W = 9 are the number of de-grees of freedom in the respective components.The background fluid is assumed to be in chemicalequilibrium, implying that δµ bg = 0. This assump-tion removes the top row of eq. (25). The remainingtwo rows determine δτ bg and δv bg in terms of q W and q t , (cid:126)q (cid:48) bg = − A − bg (Γ bg , W (cid:126)q W + Γ bg ,t (cid:126)q t ) (34)where A − bg denotes the matrix where the bottomright block of A bg is inverted and the rest of thematrix elements are zero.Equations (23) and (24) can then be expressed in6 × A(cid:126)q (cid:48) + Γ (cid:126)q = S (35) As we were completing this work, an improved numericalevaluation of these collision rates was presented in ref. [22]. Wt T W T t T bg v W v t v bg z (1/GeV) FIG. 4. Example of the solutions for the fluid perturbations, for a model with m s = 63 GeV, w = 130 GeV,and λ hs = 0 . v w = 0 .
297 and background wall shape h ( z ) = ( h / z/L w ) + 1] where h = 209 GeV and L w = 0 . − . with A ≡ γ (cid:20) A W A t (cid:21) , (cid:126)q ≡ (cid:20) (cid:126)q W (cid:126)q t (cid:21) , S ≡ γ (cid:20) S W S t (cid:21) (36)andΓ ≡ (cid:20) Γ W
00 Γ t (cid:21) − (cid:20) A W A − Γ bg , W A W A − Γ bg ,t A t A − Γ bg , W A t A − Γ bg ,t (cid:21) (37)The factors of γ = 1 / (cid:112) − v w are from Lorentzboosting to the rest frame of the wall.The W and t fluid perturbations are determinedby solving eq. (35) using the relaxation method asdescribed in ref. [23], since shooting tends to beunstable. The background fluid perturbations arefound by integrating eq. (34). One can carry outthis procedure for given values of the wall velocityand shape, and from the ensuing perturbations com-pute the friction term in the Higgs field equation ofmotion (18) using (cid:90) d p (2 π ) E δf i ( (cid:126)p, z ) ∼ = (cid:90) d p (2 π ) E f (cid:48) ,i ( (cid:126)p, z ) δ i ( z )= T (cid:104) c i ( z ) δµ i ( z ) + c i ( z ) (cid:0) δτ i ( z ) + δτ bg ( z ) (cid:1)(cid:105) (38)An example of the solutions for the perturbations isshown in fig. 4. IV.4. Solving the Equations of Motion
With the friction calculated in eq. (38), the equa-tions of motion that must be solved to determine v w and the shape of the wall are − h (cid:48)(cid:48) ( z ) + ∂V eff ( h, s, T + ) ∂h (39)+ n t T + dm t dh (cid:2) c t δµ t + c t ( δτ t + δτ bg ) (cid:3) + n W T + dm W dh (cid:2) c W δµ W + c W ( δτ W + δτ bg ) (cid:3) = 0 − s (cid:48)(cid:48) ( z ) + ∂V eff ( h, s, T + ) ∂s = 0 . (40)Deep into the bubble interior, eq. (39) is notexactly satisfied once we adopt our approximationschemes for calculating the effective potential and z (1/GeV) G e V tanh fitsh(z)s(z) FIG. 5. The wall shape that solves the equation of mo-tion for the model with m s = 63 GeV, w = 130 GeV,and λ hs = 0 .
9. The dashed curves show the best fitsusing the tanh ansatz of eqs. (48 , 49). the perturbations. The Higgs’ VEV is unchangingthere, so the kinetic term is zero. Similarly the per-turbations in the W and t fluids go to zero on bothsides of the wall. This implies that the terms propor-tional to δτ bg must exactly cancel out the potentialterm. When the perturbations are determined asdescribed above and the potential term is calculatedwith the Higgs VEV that minimizes the potentialinside the bubble, the two terms do not cancel asthey should. This is due to differences in the deriva-tion of the friction terms in comparison to the ef-fective potential. Firstly, the fluid perturbations areonly determined to linear order whereas the temper-atures that go into the effective potential, T + and T − , were calculated including non-linearities in thefluid equations. This means that while in theory T + − T − = T + δτ bg , their relationship is only ap-proximate. The other cause is that the scalar fieldswere treated as massless background fields in thefriction calculation but their full contribution wasincluded in the effective potential. There are threeways to account for this inconsistency: the HiggsVEV inside the bubble can be chosen not to mini-mize the potential but instead to cancel the frictionterm, the entire friction can be scaled to cancel thepotential term but maintaining the friction shape in z , or just the background perturbation contributionto the friction can be scaled to cancel the potentialterm. We adopt the last option, which we found tobe the most conservative choice (leading to slightlylarger wall velocities). The equations of motion thatwe actually use to determine the wall dynamics thenbecome E h ≡ − h (cid:48)(cid:48) ( z ) + ∂V eff ( h, s, T + ) ∂h (41)+ n t T + dm t dh (cid:2) c t δµ t + c t ( δτ t + yδτ bg ) (cid:3) + n W T + dm W dh (cid:2) c W δµ W + c W ( δτ W + yδτ bg ) (cid:3) = 0 E s ≡ − s (cid:48)(cid:48) ( z ) + ∂V eff ( h, s, T + ) ∂s = 0 (42)where y is an O (1) parameter chosen so that theequations are satisfied for larger positive values of z .For a given value of v w , the relaxation methodcan be used to find the shapes of h ( z ) and s ( z ) thatcome closest to solving the equations of motion. Onemust then vary v w and find a complete solution tothe equations, by iterating this procedure. A rea-sonable initial guess for both v w and the wall shapeis required, leading us to solve the equations in twostages. The first part is to guess v w and the wallshape using the tanh ansatz employed in previousstudies of wall velocities [7],[15],[16]. The seconduses these as a starting point to numerically deter-mine v w and the wall shapes. The tanh ansatz in the first stage assumes thatthe Higgs profile has the form h ( z ) = v ( T − )2 (cid:18) tanh (cid:18) zL w (cid:19) + 1 (cid:19) (43)where v ( T − ) is the Higgs VEV at temperature T − and L w is the width of the wall. The friction andshape of the singlet profile are independent of eachother, so there is no need to impose a tanh ansatz for s ; rather its profile is found by numerically solvingits equation of motion. This reduces the problemto finding values of v w and L w that come closest tosolving the Higgs equation of motion .No choice of v w and L w will exactly solve eq. (41),since the true shape is not a tanh function. Insteadwe follow ref. [15] by calculating two moments of E h in eq. (41) and finding the values of v w and L w thatmake them vanish. The two moments are taken tobe E ≡ (cid:90) h (cid:48) ( z ) E h dz = 0 (44) E ≡ (cid:90) h (cid:48) ( z ) (cid:0) h ( z ) − v ( T − ) (cid:1) E h dz = 0since with this choice the Jacobian matrix ∂ ( E , E ) /∂ ( v w , L w ) is always far from being sin-gular.The first stage of the algorithm can then be sum-marized as:1. Make a guess for v w and L w
2. Calculate T + and T − for v w
3. Determine s ( z ) by solving the s equation ofmotion using the tanh ansatz for h ( z )4. Determine the shape of the friction term forthe guessed shape of h ( z )5. Calculate the moments E and E
6. Find the new guess for v w and L w by solving E i = 0.In the second stage, we aimed to relax the tanhprofile assumption for h ( z ) and to determine itsshape more exactly. Using the values of v w and L w from the first part as new initial guesses, we solvedboth h and s equations of motion simultaneously, In an alternative implementation of this initial stage, whichis also effective, we fix the path through field space as anarc passing through the saddle point, and we work with thefield equation along that path rather than giving priorityto h or s . w = 100 GeV w = 110 GeV w = 120 GeV w = 130 GeV w = 140 GeV w = 150 GeV w = 160 GeV w = 170 GeV v w λ hs m s ( G e V ) . . . . . . . . . . . . . . . . FIG. 6. Contours of the wall velocity v w in the λ hs - m s plane, with w increasing from 100 to 170 GeV in successiveplots. The white area indicates regions where no first order transition satisfying the sphaleron bound (5) was found.In the grey hatched region, strong transitions satisfying (5) exist, but no solutions with v w < c s were found. The redcontours indicate values of the singlet self-coupling, λ s , as determined by eq. (2). For each w we show only regionscontaining viable solutions for the bubble wall parameters, within the ranges specified in Eqn. (46). using relaxation. A challenge here is that the fric-tion on the wall, which is expensive to compute, de-pends on the background h ( z ) solution. To speed upthe algorithm, we recomputed the friction only af-ter several relaxation steps. This procedure leads toeventual convergence, unless the initial guess for v w is too poor. Convergence was tested by seeing howclosely the two equations of motion were satisfied,using the squared error statistic E tot = (cid:90) (cid:2) E h + E s (cid:3) dz . (45)The best value of v w was determined by varying v w in the region of the guess from step 1 as to mini-mize E tot . An example of the wall shapes that solvethe equations of motion is given in fig. 5. It demon-strates that the actual profiles can differ significantlyfrom the tanh ansatz. V. RESULTS AND DISCUSSION
A scan of the parameter space of the scalar singletmodel was performed in the ranges0 . ≤ λ hs ≤ .
563 GeV ≤ m s ≤
114 GeV , (46)100 GeV ≤ w ≤
170 GeV . We did not find viable examples for w (cid:46)
90 or (cid:38)
180 GeV. Our results indicate that this coversmost, if not all, of the parameter space of interestfor subsonic walls.We imposed the lower bound m s > m h / h → ss ) do not apply [24]. This is a mild re-striction, since for m s < m h / (cid:46) λ hs (cid:46) .
01, whichis too small to give rise to a strong phase transition.0 r ( v n T c )/( T n v c ) v w r ) FIG. 7. The dependence of wall velocity, v w , on thesupercooling parameter, r = ( v n /T n ) / ( v c /T c ). The solidcurve shows a fit to the points. r ( v n T c )/( T n v c ) T n Models without subsonic wallsModels with subsonic walls
FIG. 8. Scatter plot of nucleation temperature T n versusthe supercooling parameter r , for all models CosmoTran-sitions found to satisfy the sphaleron washout condition. V.1. Wall Velocity Results
Our determinations of the wall speed over the fullparameter space are illustrated in in fig. 6, show-ing contours of v w in the plane of m s versus λ hs ,for a series of w values. The grey hatched regionsindicate parameters for which no transitions withsubsonic walls were found. One can see that modelswith heavier singlets and larger λ hs couplings tendto produce faster-moving walls. Generally we find aminimum value for v w , which depends on w and issmallest for w ∼
120 GeV, where the lowest speed v w ∼ = 0 .
22 is found. The parameters specifying a fewbenchmark models and their resulting phase transi-tion properties are shown in table I.Since it is numerically expensive to compute v w L h (1/GeV) L h T + T + = 120 GeV FIG. 9. The Higgs wall width L h , in units of the in-verse wall temperature T − , versus the same quantity inGeV − units. The solid line shows a fit to the points, cor-responding to the mean wall temperature T + = 120 GeV. for a given model from first principles, it is usefulto look for relations between it and other quantitiescharacterizing the strength of the phase transition,that are easier to compute. In fact we observe astrong correlation between v w and the double ratio r ≡ v n /T n v c /T c (47)where v/T is evaluated respectively at the nucleationand the critical temperatures. This is a measure ofthe degree of supercooling, and its correlation with v w is plotted in fig. 7, showing that v w increasesrapidly with r −
1. We find an analytic fit v w ∼ =0 .
53 (1 − . r − ), with deviations of order ± . r found for subsonic bubblewalls was r ∼ = 1 .
15. It remains close to unity even forstrong transitions, validating the assumption madein section IV.1 that the equations of state at thenucleation and critical temperatures do not differsignificantly from each other.The fact that a cutoff on r exists, above which it isunlikely to produce subsonic walls, can be seen in fig.8, which shows all the models tested, including those m s λ hs w v w T n T c v n /T n r
63 0.9 130 0.297 103.592 104.865 2.02 1.0281 1.0 110 0.336 124.301 125.425 1.40 1.0366 0.3 160 0.492 130.532 132.677 1.28 1.05105 0.8 110 0.530 130.646 134.461 1.24 1.11TABLE I. Benchmark models with successively fastermoving walls. Masses and temperatures are in GeV. r isthe measure of supercooling defined in eq. (47). w = 100 GeV w = 110 GeV w = 120 GeV w = 130 GeV w = 140 GeV w = 150 GeV w = 160 GeV w = 170 GeV L h T + λ hs m s ( G e V ) w = 100 GeV w = 110 GeV w = 120 GeV w = 130 GeV w = 140 GeV w = 150 GeV w = 160 GeV w = 170 GeV L s T + λ hs m s ( G e V ) FIG. 10. Top two rows (a): Like fig. 6, but showing the contours of the Higgs wall width L h T + where T + is thetemperature in front of the bubble wall. Bottom two rows (b): Like fig. 6, but showing the contours of the singletwall width L s T + .(a)(b) w = 100 GeV w = 110 GeV w = 120 GeV w = 130 GeV w = 140 GeV w = 150 GeV w = 160 GeV w = 170 GeV δ z T + λ hs m s ( G e V ) FIG. 11. Like fig. 6, but showing the contours of the wall separation parameter δ z , defined in eq. (49). found not to have slow bubble walls. It clearly showsthat for r (cid:38) .
1, few transitions produce subsonicwalls, whereas below that point all the models testedwere found to do so.Fig. 6 shows that subsonic walls require the sin-glet to be relatively light, m s (cid:46)
110 GeV, often witha relatively large coupling to the Higgs, λ hs ∼
1. If s is long-lived enough to escape detection within acollider, Ref. [25] suggests that a singlet with theseproperties may be a realistic target for the high-luminosity LHC (in the MET plus forward jets chan-nel, from vector-boson fusion production of an off-shell Higgs). On the other hand, if we take the modelat face value, as a complete model with a standardthermal history, Ref. [25] also finds that the LUXdirect detection experiment [26] rules out m s (cid:46) s would make a subdominant con-tribution to the dark matter. Of course, additionalmodel ingredients can easily make s unstable on cos-mological time scales without affecting our phase-transition and wall-velocity results. V.2. Wall Shape Results
Although Fig. 5 shows that the wall shapes de-viate from a tanh profile, it is nevertheless a use-ful approximation for concisely encoding informa-tion about the wall shapes. We have accordinglyanalyzed our results from the fully numerical algo-rithm to find the best-fit tanh profiles, including apossible offset δ z between the Higgs and the singletprofiles: h fit = h (cid:20) (cid:18) zL h (cid:19)(cid:21) (48) s fit = s (cid:20) − tanh (cid:18) z − δ z L s (cid:19)(cid:21) , (49)where we have allowed for independent widths L h and L s of the Higgs and singlet profiles.To display results for the wall thicknesses, we haveopted to use dimensionless combinations like L h T + ,where T + is the temperature of the wall. If one wantsto translate these into absolute thicknesses, it can bedone using the strong correlation between L h T + and L h in GeV − units, shown in Fig. 9. Since all modelswith subsonic walls have nucleation temperatures in3 v w L h ( / G e V ) FIG. 12. The dependence of Higgs wall width, L h , onthe wall velocity, v w . the range 80 GeV ≤ T n ≤
140 GeV (see Fig. 8), andfor slow walls the wall temperature does not deviatemuch from the nucleation temperature, the relation-ship between these two ways of characterizing L h islinear with relatively little scatter: L h T + ∼ = L h T + = 120 GeV arerelatively small.Contour plots of L h T + , L s T + , and δ z T + similarto those for v w are presented in Figs. 10 and 11.We find that faster walls tend to be thinner andhave smaller offsets. These relationships are plot-ted in Figs. 12-14, which show strong correlations,especially in the case of L h . With rare exceptions, L s < L h , with L s typically smaller than L h by 20-30%.As discussed in Appendix C, a small number ofpoints were found to have extra potential minima orplateaus in the middle of the wall. The results forthese points are likely to change with a more carefultreatment of the effective potential. However, thesepoints are rare, and including them does not changeany of our conclusions. VI. CONCLUSION
This work has laid out a more quantitativemethodology than has been previously used, for cal-culating the wall velocity of bubbles during the elec-troweak phase transition with an additional scalarfield. We improved on previous similar studies bysolving for the actual profiles of the scalar fields,rather than just parametrizing them using a tanhansatz. Other improvements made here include useof the one-loop Coleman-Weinberg contributions tothe potential including the effect of thermal masses, accounting for the sphericity of the bubbles, not ex-panding fluid perturbations to first order in m/T ,and performing a scan over the three-dimensionalparameter space.Scanning over the parameter space reveals thatthe scalar singlet model is able to produce slow bub-ble walls that are preferable for electroweak baryo-gensis to occur, down to a minimum wall velocityof v w ∼ = 0 .
22. These examples of slow-moving wallsonly occur in phase transitions with small amountsof supercooling.There are a few ways in which this study can beextended by future work. The precision of the wallvelocity calculation could be improved by includingadditional sources of friction such as from the scalarfields and IR gauge boson modes. For a completeanalysis of electroweak baryogensis in the Z scalarsinglet model, this analysis could be embedded in amore complete model that includes a new source ofCP-violation in order to determine the size of the v w L s ( / G e V ) FIG. 13. Like fig. 12 but for L s . v w z ( / G e V ) FIG. 14. Like fig. 12 but for δ z . ACKNOWLEDGMENTS
We thank B. Laurent for useful discussions andcomments on the draft. JC and DTS thank the As-pen Center for Physics for providing a stimulatingenvironment where this work was initiated. Thecomputations in this work were run using equip-ment funded by the Canada Foundation for Inno-vation and supported by the Centre for AdvancedComputing at Queen’s University. Besides the pre-viously mentioned CosmoTransitions, the code usedfor the calculations utilised Eigen [27] and the GNUScientific Library [28]. AF and JC are supported byNSERC (Natural Sciences and Engineering ResearchCouncil, Canada).
Appendix A: Effective potential
The one-loop contribution to the potential can beapproximated as V = (cid:88) i = h,s,χ,t,W,Z,γ n i m i ( h, s, T )64 π (cid:20) ln (cid:18) m i ( h, s, T ) v (cid:19) − c i (cid:21) (A1)where n i is the number of degrees of freedom of eachparticle. For the scalar fields, longitudinal W/Z andtop quark c i = 3 / c i = 1 /
2, in the MS scheme. The top quarkis the only fermion included in the sum since thecontributions from lighter fermions are suppressedby their small Yukawa couplings. χ stands for theGoldstone boson contributions.The one-loop contribution acquires a temperaturedependence through the thermal masses of the par-ticles, in this method of carrying out the ring resum-mation [29]. It has been shown that for sufficientlystrong phase transitions, a more careful treatmentof thermal masses can be important [30].The scalar masses in eq. (A1) are given by theeigenvalues of the mass matrix: M ,ij ≡ ∂ V∂φ i ∂φ j + m T,i δ ij (A2)where φ i and φ j are the five scalar fields summed over in eq. (A1)and m T,h = T (cid:18) g + g (cid:48)
16 + y t λ h λ hs (cid:19) (A3) m T,χ = m T,h (A4) m T,s = T (cid:18) λ hs λ s (cid:19) . (A5)The three mass eigenvalues associated with theGoldstone bosons vanish in the vacuum state makingthose terms in eq. (A1) formally divergent. This isproperly dealt with by introducing a scale coincidingwith the Higgs mass, m h , to cut off the IR divergence[31].The masses associated with the longitudinalmodes of the gauge bosons in eq. (A1) are given bythe eigenvalues of the mass matrix: M ,ij ≡ g h g h g h gg (cid:48) h gg (cid:48) h g (cid:48) h + 116 T diag( g , g , g , g (cid:48) ) (A6)The rest of the field-dependent masses in eq. (A1)are given by: m ,w = g h m ,z = ( g + g (cid:48) ) h m ,γ = 0 m t = y t h V CT = 12 δm h h + 12 δm s s + 14 δλ h h + 14 δλ s s + 14 δλ hs h s (A8)The five counterterms were chosen to ensure that thefull effective potential at T = 0 maintains its tree-level values for the scalar masses, potential minima,and scalar mixing. This is done by imposing thefollowing conditions at T = 0: ∂V∂h (cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 = ∂V∂s (cid:12)(cid:12)(cid:12)(cid:12) h =0 ,s = w = 0 (A9) ∂ V∂h (cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 = m h , ∂ V∂s (cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 = m s (A10)5and ∂ V∂h ∂s (cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 = λ hs (A11)where m s = (cid:113) λ hs v − λ s w is the mass of thescalar singlet in the true vacuum.The resulting counterterm parameters are foundto be δm h = (cid:18) ∂ V ∂h − v ∂V ∂h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 (A12) δm s = (cid:18) − ∂ V ∂s + v ∂ V ∂h ∂s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 (A13) δλ h = 12 v (cid:18) v ∂V ∂h − ∂ V ∂h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 (A14) δλ s = − δm s w − w ∂V ∂s (cid:12)(cid:12)(cid:12)(cid:12) h =0 ,s = w (A15)and δλ hs = − ∂ V ∂h ∂s (cid:12)(cid:12)(cid:12)(cid:12) h = v ,s =0 (A16)Lastly, the temperature dependence of the poten-tial is given by V T = − T π J F (cid:18) m t ( h ) T (cid:19) + (cid:88) i = h,s,χ,W,Z n i T π J B (cid:18) m i ( h, s, T ) T (cid:19) (A17)where J F and J B are functions which describefermions and bosons temperature-dependent contri-bution to the one-loop potential. The functions arecalculated from J F ( y ) = (cid:90) ∞ x ln (cid:16) e − √ x + y (cid:17) dx (A18)and J B ( y ) = (cid:90) ∞ x ln (cid:16) − e − √ x + y (cid:17) dx (A19)These equations fully describe the one-loop poten-tial of the scalar fields. Appendix B: Linearized Boltzmann Equations
The following derivation of the linearized mo-ments to the Boltzmann equation, which are used toto determine the friction of the equation of motion,follows closely to that originally expressed in [7].The difference between that derivation and the onehere is that the full dependence of m/T is includedhere instead of expanding to lowest order. This al-lows for stronger phase transitions to be quantita-tively studied.As noted in eqs. (20-22) the fluids are describedby the distribution function f i ( E, z ) = 1 e E + δ i ( z )) /T ± / − is for fermions/bosons and δ i ( z ) = − (cid:104) T ( δµ i + δµ bg )( z ) + E ( δτ i + δτ bg )( z )+ p z ( δv i + δv bg )( z ) (cid:105) (B2)The background fluid is in chemical equillibrium sofor the rest of the derivation δµ bg = 0 Deviationsfrom equilibrium in the fluids are governed by theBoltzmann equation df i dt = − C [ f i ( E, z )] (B3)The left side of eq. (B3) can be expanded as df i dt = f (cid:48) ,i (cid:18) dEdt + dδ i dt (cid:19) (B4)where f (cid:48) ,i ≡ ∂ E f i | δ i =0 (B5)In the fluid’s reference frame dδ i dt = ∂ t δ i + p z E ∂ z δ i − ( m i ) (cid:48) E ∂ p z δ i (B6)Starting with the last term − ( m i ) (cid:48) E ∂ p z δ i = ( m i ) (cid:48) E ( δv i + δv bg ) (B7)As will be shown the perturbations are sourced by aterm proportional to ( m i ) (cid:48) E f (cid:48) ,i so terms like the oneabove which are proportional to ( m i ) (cid:48) E f (cid:48) ,i δ i are onthe same order as δ i and therefore are ignored tolinear order.This may raise the concern that if m i /T is notsmall and δ i ∝ ( m i ) (cid:48) E , does the linear approxima-tion break down? The tanh ansatz can be used to6set a rough condition on the relation between v n /T n and LT under which taking the linear order is valid.That will be derived at the end of this section.Next one observes that ∂ t = v w ∂ z in the fluid’sreference frame, so to linear order in the perturba-tions dδ i dt = (cid:16) v w + p z E (cid:17) ∂ z δ i . (B8)Going back to eq. (B4), the term independent of δ i acts as the source term in the perturbations equa-tions. dEdt = ddt ( p + m i ) / = 12( p + m i ) / dm i dt (B9)= v w ( m i ) (cid:48) E Therefore the Boltzmann equation becomes f (cid:48) ,i (cid:16) v w + p z E (cid:17) ∂ z δ i + C [ f i ] = − v w f (cid:48) ,i ( m i ) (cid:48) E (B10)which when expanding out δ i it becomes − f (cid:48) ,i ( v w + p z E )[ T δµ (cid:48) i + E ( δτ (cid:48) i + δτ (cid:48) bg ) + p z ( δv (cid:48) i + δv (cid:48) bg )]+ C [ f i ] = − v w f (cid:48) ,i ( m i ) (cid:48) E (B11)Three moments are taken to turn this into a sys-tem of ordinary differential equations. The threemoments are (cid:82) d p (2 π ) , (cid:82) ET d p (2 π ) , and (cid:82) p z d p (2 π ) .When taking the first moment, all terms propor-tional to p z integrate to zero leaving (cid:90) d p (2 π ) (cid:18) − f (cid:48) ,i v w [ T δµ (cid:48) i + E ( δτ (cid:48) i + δτ (cid:48) bg )] − f (cid:48) ,i p z E ( δv (cid:48) i + δv (cid:48) bg ) + C [ f i ] (cid:19) = (cid:90) d p (2 π ) ]) (cid:18) − v w f (cid:48) ,i ( m i ) (cid:48) E (cid:19) (B12)Two sets of variabels are then introduced. c ij = − (cid:90) f (cid:48) ,i E j − T j +1 d p (2 π ) (B13)and d ij = − (cid:90) f (cid:48) ,i p E j − T j +1 d p (2 π ) (B14) After noting that p z = p / T v w c i δµ (cid:48) i + T v w c i ( δτ (cid:48) i + δτ (cid:48) bg )+ T v w d i ( δv (cid:48) i + δv (cid:48) bg ) / (cid:90) d p (2 π ) C [ f i ]= T v w c i ( m i ) (cid:48) T v w c i δµ (cid:48) i + v w c i ( δτ (cid:48) i + δτ (cid:48) bg )+ v w d i ( δv (cid:48) i + δv (cid:48) bg ) / (cid:90) d p (2 π ) C [ f i ] T = v w c i ( m i ) (cid:48) T (B16)The second moment equation is the exact sameexcept with an extra factor of E/T in each termleading to v w c i δµ (cid:48) i + v w c i ( δτ (cid:48) i + δτ (cid:48) bg )+ v w d i ( δv (cid:48) i + δv (cid:48) bg ) / (cid:90) d p (2 π ) E C [ f i ] T = v w c i ( m i ) (cid:48) T (B17)For the third moment equation, due to the extrafactor of p z , the opposite set of terms in eq. (B11)compared to the first two moments integrates to zeroleaving (cid:90) d p (2 π ) (cid:18) − f (cid:48) ,i p z E [ T δµ (cid:48) i + E ( δτ (cid:48) i + δτ (cid:48) bg )] − f (cid:48) ,i v w p z ( δv (cid:48) i + δv (cid:48) bg ) (cid:19) = 0 (B18)which becomes d i δµ (cid:48) i / d i ( δτ (cid:48) + δτ (cid:48) bg ) / v w d i ( δv (cid:48) i + δv (cid:48) bg ) / A i ( (cid:126)q i + (cid:126)q bg ) (cid:48) + Γ i (cid:126)q i = S i (B20)with A i , Γ i , S i , and q i all taking the same form asthey do in section IV.Perturbations, q i , are sourced by a term propor-tional to ( m i ) (cid:48) T so if ( m i ) (cid:48) T ∼ h ( z ) = v (cid:18) tanh( zLT ) + 1 (cid:19) (B21)7 h (GeV) s ( G e V ) z (1/GeV) V ( G e V ) (a) (b)FIG. 15. Left (a): Contours of V eff for the model with λ hs = 1 . w = 120 GeV, and m s = 69 GeV, evaluated at thewall temperature of T + = 114 .
075 GeV. Right (b): The potential for this model projected on the shape of the wall. z (1/GeV) V ( G e V ) FIG. 16. The scalar potential along the wall for themodel with λ hs = 0 . w = 110 GeV, and m s = 81 GeVevaluated at the wall temperature of T + = 126 .
130 GeV.
This conditions will first break down with the topquark which has a mass given by m t ( z ) /T = y t h ( z ) √ T (B22)Then by taking the derivative( m t ) (cid:48) T = ( vT ) y t sech ( zLT ) (cid:18) tanh( zLT ) + 1 (cid:19) LT (B23)At its maximum value this is equal to( m t ) (cid:48) T (cid:12)(cid:12)(cid:12)(cid:12) max = 4( vT ) y t LT (B24) z (1/GeV) G e V tanh fitsh(z)s(z) FIG. 17. The shape of the scalar fields for the modelthe model with λ hs = 1 . w = 120 GeV, and m s = 69GeV. By ensuring that ( m t ) (cid:48) T (cid:12)(cid:12) max < (cid:18) vT (cid:19) < . LT (B25)This condition is easily met by all the wall foundto have subsonic walls in this paper therefore indi-cating that the linear order approximation is valid. Appendix C: Questionable Transitions
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