Sphalerons and the Electroweak Phase Transition in Models with Higher Scalar Representations
aa r X i v : . [ h e p - ph ] N ov Prepared for submission to JHEP
Sphalerons and the Electroweak PhaseTransition in Models with Higher ScalarRepresentations
Amine Ahriche a,b,c
Talal Ahmed Chowdhury d and Salah Nasri ea Department of Physics, University of Jijel, PB 98 Ouled Aissa, DZ-18000 Jijel, Algeria b Fakult¨at f¨ur Physik, Universit¨at Bielefeld, 33501 Bielefeld, Germany c The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34014, Trieste, Italy d SISSA, Via Bonomea 265, 34136, Trieste, Italy e Department of Physics, UAE University, P.O. Box 17551, Al-Ain, United Arab Emirates
E-mail: [email protected], [email protected], [email protected]
Abstract:
In this work we investigate the sphaleron solution in a SU (2) × U (1) X gauge theory, which also encompasses the Standard Model, with higher scalar rep-resentation(s) ( J ( i ) , X ( i ) ). We show that the field profiles describing the sphaleronin higher scalar multiplet, have similar trends like the doublet case with respect tothe radial distance. We compute the sphaleron energy and find that it scales linearlywith the vacuum expectation value of the scalar field and its slope depends on therepresentation. We also investigate the effect of U (1) gauge field and find that it issmall for the physical value of the mixing angle, θ W and resembles the case for thedoublet. For higher representations, we show that the criterion for strong first orderphase transition, v c /T c > η , is relaxed with respect to the doublet case, i.e. η < Keywords: sphalerons, scalar multiplets. ontents U (1) X Field and the Sphaleron Energy 94 Sphaleron Decoupling Condition 125 Conclusion 15A Asymptotic solutions 15
In the Standard Model (SM), the anomalous baryonic and leptonic currents lead tofermion number non-conservation due to the instanton induced transitions betweentopologically distinct vacua of SU (2) gauge fields [1, 2] and at zero temperature, therate is of the order, e − π/α w , α w ∼ /
30, which is irrelevant for any physical phenom-ena. However, there exists a static unstable solution of the field equations, known assphaleron [3–6], that represents the top of the energy barrier between two distinctvacua and at finite temperature, because of thermal fluctuations of fields, fermionnumber violating vacuum to vacuum transitions can occur which are only suppressedby a Boltzmann factor, containing the height of the barrier at the given tempera-ture, i.e. the energy of the sphaleron [7]. Such baryon number violation induced bythe sphaleron is one of the essential ingredients of Electroweak Baryogenesis [8–13]and therefore it has been extensively studied not only in the SM [14–24] and butalso in extended SM variants such as, SM with a singlet [25, 26], two Higgs dou-blet model [27], Minimal Supersymmetric Standard Model [28], the next-to-MinimalSupersymmetric Standard Model [29] and 5-dimensional model [30].As many SM extensions involve non-minimal scalar sectors, it is instructive todetermine the behavior of the sphaleron for general SU (2) scalar representations.Although, apart from some exceptions like Georgi-Machacek [31] and isospin-3 mod-els [32], large Higgs multiplets other than the doublet are stringently constrained– 1 –y electroweak precision observables. In addition, the presence of scalar multipletswith isospin J ≥ landau ≤
10 TeV [33]. Moreover as shown in [34, 35], by saturating unitarity boundon zeroth order partial wave amplitude for the 2 → SU (2) multiplet tohave isospin J ≤ / J ≤
4. Therefore it can be seen thatlarge scalar representations of SM gauge group are generally disfavored.Still, motivated by the dark matter content and baryon asymmetry of the uni-verse, one can assume a hidden or dark sector with its own gauge interactions. If theinteraction between SM and hidden sector is feeble in nature, they may not equili-brate in the whole course of the universe. Therefore, the hidden sector can be fairlyunconstrained apart from its total degrees of freedom such that the sector doesn’tchange the total energy density of the universe in such way that the universe had amodified expansion rate in earlier times, specially at the BBN and CMB era. Withthis possibility in mind, we can consider the hidden sector to have SM-like gaugestructure that contains scalar multiplets larger than doublet and also has its ownspontaneous symmetry breaking scale (the possibility of non-abelian gauge structurein dark sector and non-SM sphaleron in symmetric phase for such models are alsoaddressed in [36, 37]). For this reason, it is interesting to ask what could be the na-ture of the sphaleron in such SM-like SU (2) × U (1) X gauge group with general scalarmultiplets. Furthermore, as sphaleron is linked with nontrivial vacuum structure ofnon-abelian gauge theory, it is relevant to see the effect of large scalar multiplets inhot gauge theories.This paper is organized as follows. In section 2 we discuss the spherically sym-metric ansatz for larger scalar multiplets and consequently calculated the energyfunctional and variational equations for scalar multiplet ( J, X ), give different numer-ical results. In section 3 we investigate the effect of U (1) X field on sphaleron energyand study the sphaleron energy dependence on the scalar vev. Section 4 is devoted tothe conditions of the sphaleron decoupling during the electroweak phase transition,and in section 5 we conclude. In Appendix A, we have presented the asymptoticsolutions and their dependence on the representation ( J, X ). The standard way to find sphaleron solution in the Yang-Mills-Higgs theory is toconstruct non-contractible loops in field space [5]. As the sphaleron is a saddle pointsolution of the configuration space, it is really hard to find them by solving thefull set of equations of motion. Instead one starts from an ansatz depending on aparameter µ that characterizes the non-contractible loop in the configuration space– 2 –nd corresponds to the vacuum for µ = 0 and π while µ = π corresponds the highestenergy configuration, in other words, the sphaleron.Consider the scalar multiplet Q , charged under SU (2) × U (1) X group, is in J representation and has U (1) X charge X . Here SU (2) and U (1) X can be applicable forboth standard model gauge group or SM-like gauge group of the hidden sector. Thegenerators in this representation are denoted as J a such that, T r [ J a J b ] = D ( R ) δ ab where D ( R ) is the Dynkin index for the representation. As our focus is on the SM,we define the charge operator, ˆ Q c = J + X and require the neutral component( J = − X ) of the multiplet to have the vacuum expectation value (vev).The gauge-scalar sector of the Lagrangian is L = − F aµν F aµν − f µν f µν + ( D µ Q ) † D µ Q − V ( Q ) , (2.1)with scalar potential V ( Q ) = − µ Q Q † Q + λ ( Q † Q ) + λ ( Q † J a Q ) . (2.2)It was shown in [26] that the kinetic term of the scalar field makes larger contributionto the sphaleron energy than the potential term. Therefore, for simplicity, we haveconsidered CP-invariant scalar potential involving single scalar representation to de-termine the sphaleron solution. It is straightforward to generalize the calculation forthe potential with multiple scalar fields .Also for convenience we elaborate, F aµν = ∂ µ A aν − ∂ ν A aµ + gǫ abc A bµ A cν ,f µν = ∂ µ a ν − ∂ ν a µ ,D µ Q = ∂ µ Q − igA aµ J a Q − ig ′ a µ XQ, (2.3)where, g and g ′ are the SU (2) and U (1) X gauge couplings. The mixing angle θ W istan θ W = g ′ /g .The scalar sector plays an essential role in constructing sphaleron and the sym-metry features of the ansatz partly depends on the SU (2) representation and U (1) X charge assignment of the scalar that acquires a vev. The simplest possibility is toconsider a spherically symmetric ansatz because spherical symmetry enables one tocalculate the solution and the energy of the sphaleron without resorting into fullpartial differential equations. Therefore one may ask, which scalar representationimmediately allows the spherical symmetric ansatz.As pointed out in [16], spherically symmetric configurations are those for whichan O (3) rotation of spatial directions are compensated by the combination of SU (2)gauge and SU (2) global transformation. The existence of this SU (2) global symmetry In fact, in the SM, one needs large couplings between Higgs and extra scalars to trigger a strongfirst order phase transition. – 3 –s manifest for the Higgs doublet as the potential for the doublet has SO (4) ∼ SU (2) × SU (2) global symmetry which is broken by the scalar vev to SU (2) ∼ SO (3)symmetry that leads to the mass degeneracy of three gauge bosons of SU (2). Onecan immediately see that this degeneracy will be lifted when the U (1) X is turned on.Following the same reasoning, one can find other scalar multiplets that will lead tomass degeneracy of A aµ ’s in SU (2) gauge theory after the symmetry is broken.In the case of many scalar representations Q ( i ) with J ( i ) and charge X ( i ) , thecorresponding vev’s are h Q ( i ) i = v i √ (0 , .., , .., T , where the non-zero neutral com-ponent quantum numbers are ( J ( i ) , J ( i )3 = − X ( i ) ). Now from the scalar kinetic term, L ⊃ g X i h Q ( i ) † i J ( i ) a J ( i ) b h Q ( i ) i A aµ A µb = g X i v i ( J ( i ) ( J ( i ) + 1) − X ( i )2 ) A + µ A µ − + g X i v i X ( i )2 A µ A µ . (2.4)where A ± µ = A µ ∓ iA µ . So the condition for having equal coupling of three gaugefields to the neutral component leads to the tree-level condition ρ = P i v i ( J ( i ) ( J ( i ) + 1) − X ( i )2 )2 P i v i X ( i )2 = 1 . (2.5)In the case of one scalar multiplet, this can be reduced to J ( J + 1) = 3 X . Themultiplets satisfying the above condition are ( J, X ) = ( , ) , (3 , .. . Intuitively, onecan consider that the scalar multiplet enables the three gauge fields to scale uniformlylike a sphere in a three dimensional space. In the following we will address the energy functional and the variational equationsof the sphaleron. The classical finite energy configuration are considered in a gaugewhere the time component of the gauge fields are set to zero. Therefore the classicalenergy functional over the configuration is E ( A ai , a i , Q ) = Z d x (cid:20) F aij F aij + 14 f ij f ij + ( D i Q ) † ( D i Q ) + V ( Q ) (cid:21) . (2.6)The non-contractible loop (NCL) in configuration space is defined as map S × S ∼ S into SU (2) ∼ S using the following matrix U ∞ ∈ SU (2) [19], U ∞ ( µ, θ, φ ) = (cos µ + sin µ cos θ ) I + i sin 2 µ (1 − cos θ ) τ + 2 i sin µ sin θ (sin φτ + cos φτ ) , (2.7)– 4 –here µ is the parameter of the NCL and θ , φ are the coordinates of the sphere atinfinity. Also, τ a are the SU (2) generators in the fundamental representation. Wealso define the following 1-form i ( U ∞− ) dU ∞ = X a F a τ a , (2.8)which gives F = − [2 sin µ cos( µ − φ ) − sin 2 µ cos θ sin( µ − φ )] dθ − [sin 2 µ cos( µ − φ ) sin θ + sin µ sin 2 θ sin( µ − φ )] dφ,F = − [2 sin µ sin( µ − φ ) + sin 2 µ cos θ cos( µ − φ )] dθ + [sin µ sin 2 θ cos( µ − φ ) − sin 2 µ sin θ sin( µ − φ )] dφ,F = − sin 2 µ sin θdθ + 2 sin θ sin µdφ. (2.9)As shown in [19], the NCL starts and ends at the vacuum and consists of threephases such that in first phase µ ∈ [ − π ,
0] it excites the scalar configuration, in thesecond phase µ ∈ [0 , π ] it builds up and destroys the gauge configuration and in thethird phase µ ∈ [ π, π ] it destroys the scalar configuration.The field configurations in the first and third phases, µ ∈ [ − π ,
0] and µ ∈ [ π, π ]are gA ai τ a dx i = g ′ a i dx i = 0 , (2.10)and Q = v (sin µ + h ( ξ ) cos µ ) √ (cid:0) .. .. (cid:1) T , (2.11)with ξ = g Ω r is radial dimensionless coordinate and Ω is the mass parameter usedto scale r − , which we choose in what follows as Ω = m W /g . In the second phase µ ∈ [0 , π ], the field configurations are gA ai τ a dx i = (1 − f ( ξ ))( F τ + F τ ) + (1 − f ( ξ )) F τ ,g ′ a i dx i = (1 − f ( ξ )) F , (2.12)and Q = vh ( ξ ) √ (cid:0) .. .. (cid:1) T . (2.13)Here, f ( ξ ), f ( ξ ), f ( ξ ) and h ( ξ ) are the radial profile functions. From Eq.(2.12),one can see that in the spherical coordinate system, for the chosen ansatz, the gaugefixing has led to, A ar = a r = a θ = 0. Moreover, similar to Eq.(2.12), the gauge fieldsacting on the scalar field Q can be written as gA ai J a dx i = (1 − f )( F J + F J ) + (1 − f ) F J . (2.14)Finally the energy over the NCL for the first and third phases is,– 5 – ( h, µ ) = 4 π Ω g Z ∞ dξ (cid:20) cos µ v Ω ξ h ′ + ξ V ( h, µ ) g Ω (cid:21) , (2.15)and for second phase, E ( µ, f, f , f , h ) = 4 π Ω g Z ∞ dξ (cid:20) sin µ ( 83 f ′ + 43 f ′ ) + 8 ξ sin µ { f (1 − f ) + 13 { f (2 − f ) − f } } + 43 ( gg ′ ) { sin µf ′ + 2 ξ sin µ (1 − f ) } + v Ω { ξ h ′ + 43 sin µh { ( J ( J + 1) − J )(1 − f ) + J ( f − f ) }} + ξ g Ω V ( h ) (cid:21) . (2.16)From Eq.(2.16), the maximal energy is attained at µ = π which corresponds to thesphaleron configuration.If there are multiple representations J ( i ) with non-zero neutral components J ( i )3 , Q ( i ) = v i h i ( ξ ) √ (0 , .., .., T , the energy of the sphaleron can be parameterized as E sph = E ( µ = π π Ω g Z ∞ dξ [ (cid:20) f ′ + 43 f ′ + 83 ξ { f (1 − f ) + ( f (2 − f ) − f ) } + 43 ( gg ′ ) { f ′ + 2 ξ (1 − f ) } + X i { v i Ω ξ h ′ i + 43 h i [2 α i (1 − f ) + β i ( f − f ) ] } + ξ V ( v i h i ) g Ω (cid:21) , (2.17)where the parameters α i = ( J ( i ) ( J ( i ) + 1) − J ( i )23 ) v i , β i = J ( i )23 v i Ω , (2.18)refer to the scalar field couplings to the charged and neutral gauge fields respectively.The energy functional, Eq.(2.17) will be minimized by the solutions of the fol-lowing variational equations f ′′ + 2 ξ (1 − f )[ f ( f −
2) + f (1 + f )] + X i α i h i (1 − f ) = 0 ,f ′′ − ξ [3 f + f ( f − f )] + X i β i h i ( f − f ) = 0 ,f ′′ + 2 ξ (1 − f ) − g ′ g X i β i h i ( f − f ) = 0 ,h ′′ i + 2 ξ h ′ i − v i ξ h i [2 α i (1 − f ) + β i ( f − f ) ] − g v i Ω ∂∂φ i V ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ k = v k h k = 0 , (2.19)– 6 –ith the boundary conditions for Eq.(2.19) are given by: f (0) = f (0) = h (0) = 0, f (0) = 1 and f ( ∞ ) = f ( ∞ ) = f ( ∞ ) = h i ( ∞ ) = 1. For g ′ →
0, we have, f ( ξ ) → f ( ξ ) → f ( ξ ). The behavior ofthe field profiles Eq.(2.19) at the limits ξ → ξ → ∞ are shown in Appendix A.According to the last term in both first and second lines in Eq.(2.19), it seems thatthe couplings of the scalar to gauge components, i.e. Eq.(2.18) will play the mostimportant role in the profile’s shape as well as in the sphaleron energy. The equalitybetween the parameters α i and β i leads to the case Eq.(2.5) and any differencebetween α i and β i will characterize a splitting between the functions f and f , andtherefore a departure from the spherical ansatz that was defined in [5]. Here we are interested in investigating the properties of the field profiles for differentscalar representations and vevs. First we have studied the field profiles for only SU (2) with scalar representation ( J, X ) where g ′ is taken to be zero and consequently f →
1. The scalar representations are taken as (
J, X ) = { (1 / , / , , / , / / , / , , , } and two scalar vevs: v = 50 GeV and v =350 GeV. Here we are focusing on the sphaleron solution in a generic SU (2) × U (1) X case; therefore, we have chosen representative values of the vev which also containthe SM case, v = 246 GeV within the range. Moreover, for each representation,the quartic coupling is set to be 0.12 and the mass parameter µ Q is determined bycoupling and the scalar vev. For this parameter set, the mass of the scalar fieldremains smaller than 12 m W so there is no appearance of bisphalerons in our case.The field profiles are given in Figure 1.According to Figure 1, one can make the following remarks: • Comparing the cases of small vev, v = 50 GeV and large vev, v = 350 GeV,it can be seen that all field profiles tend quickly to the unity as the vev getslarger. This could explain the dependence of sphaleron energy Eq.(2.17) on thescalar vev. • When the scalar representation is large (large J so that large α ), the profilefor charged gauge field (i.e., f ( ξ )) tends to 1 faster with ξ , in contrast with thescalar field profile, h ( ξ ). • For the neutral gauge field profile f ( ξ ), it is identical to f ( ξ ) for the represen-tation (1 / , /
2) because it satisfies ρ = 1 (or J ( J + 1) = 3 X ) condition. • For the same value of the vev and the isospin J , the field profile f ( ξ ) tends to1 faster for larger values of J , i.e. larger values of β . • The scalar field profiles h ( ξ ) seem to be not sensitive to the values of J .– 7 – f ( ξ ) ξ =m W r υ =50 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 f ( ξ ) ξ =m W r υ =350 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 f ( ξ ) ξ =m W r υ =50 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 f ( ξ ) ξ =m W r υ =350 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 h ( ξ ) ξ =m W r υ =50 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 h ( ξ ) ξ =m W r υ =350 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) Figure 1 . The field profiles f ( ξ ), f ( ξ ) and h ( ξ ) as the function of the radial coordinate.In the left figures, we set the vacuum expectation value to be v = 50 GeV and in the right,it’s v = 350 GeV. Therefore, it is seen that the gauge field profiles tend to unity faster in contrastto the scalar field profiles with radial coordinate for large couplings of the scalar tocharged gauge boson, α and neutral gauge boson, β . In the next section, we will seethe impact of this feature on the sphaleron energy.– 8 – The Effect of U (1) X Field and the Sphaleron Energy
In the presence of a non-zero U (1) X gauge coupling g ′ or non-zero Weinberg angle θ W , the U (1) X gauge field will be excited and the spherical symmetry will be reducedto axial symmetry. In [22], it was shown for the SM with one Higgs doublet thatwhen the mixing angle is increased, the energy of the sphaleron decreases and itchanges the shape from a sphere at θ W = 0 to a very elongated spheroid at largemixing angle. However, for the physical value of the mixing angle, the sphalerondiffers only little from the spherical sphaleron. On the other hand, for multipletsnot satisfying Eq.(2.5), the shape of the corresponding sphaleron will be spheriodalinstead of spherically symmetric in the SU (2) case. In such cases, the large valueof the mixing angle may be significant for the energy and shape of the sphaleronfor large multiplet [38]. In the following, we have adopted the small mixing anglescenario so that SU (2) × U (1) X sphalerons are not so different than the SU (2) case;and we will work at first order of small θ W value.In Figure 2, we have presented the field profile f ( ξ ) for different values of vev( v = 50 ,
350 GeV) and different representations (
J, X ). f ( ξ ) ξ =m W r υ =50 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0 5 10 15 20 f ( ξ ) ξ =m W r υ =350 GeV (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) Figure 2 . The field profile f ( ξ ) as a function of the radial coordinate. In the left figure,we set the vacuum expectation value to be v = 50 GeV and in the right, it’s v = 350 GeV. In the case of a SU (2) × U (1) X sphaleron, we have presented only the fieldprofile f ( ξ ) since the other profiles ( f ( ξ ), f ( ξ ) and h ( ξ )) are very close to the caseof vanishing Weinberg angle shown in the previous section. In Figure 2, one cannotice that the field profile f ( ξ ) is just a deviation from unity similar to the singletscalar profile in models with singlets [26] and it gets closer to unity as the X valuesbecomes smaller and smaller. Indeed, it is exactly one for the representations (1 , ,
0) which means that in those cases the sphaleron energy is not affected bythe existence of U (1) X gauge field. – 9 –hen we have θ W = 0, even when one starts with a i = 0, the following U (1) X current j i will induce a i , j i = i g ′ [ Q † D i Q − ( D i Q ) † Q ] , (3.1)In the leading order approximation of θ W , we can neglect the a i contribution inthe covariant derivative. Therefore the non-zero component of the U (1) X current inthe chosen ansatz is [5] j φ = g ′ sin θr X i v i J ( i )3 h i (1 − f ) . (3.2)Because of induced field a i , there will be a dipole contribution to the energy, E dipole = Z d xa i j i = − π g Ω X i v i J ( i )3 Z ∞ dξ (1 − f )(1 − f ) h i , (3.3)and the sphaleron energy will be E sph | θ W =0 = E sph | θ W =0 + E dipole . (3.4)In the current Eq.(3.2) the contribution of the U (1) X gauge field is generallyneglected in the literature and when we consider it, the current and the dipole energybecome j φ = g ′ sin θr X i v i J ( i )3 h i ( f − f ) ,E ′ dipole = − π g Ω X i v i J ( i )3 Z ∞ dξ (1 − f )( f − f ) h i , (3.5)Therefore the dipole contribution Eq.(3.3) is expected to be almost equal to thedifference between Eq.(2.17) and the same quantity with g ′ = 0, i.e., E dipole ≃ ∆ E sph = E sph ( g ′ = 0) − E sph ( g ′ = 0). In order to probe this, we estimate thedifference between the sphaleron energy in the non-zero and zero mixing cases inthree different ways: (A) ∆ E sph = E sph ( g ′ = 0) − E sph ( g ′ = 0) with E sph is given inEq.(2.17); (B) ∆ E sph = E dipole with U (1) X field neglected as given in Eq.(3.3); and(C) ∆ E sph = E ′ dipole as shown in Eq.(3.5). These three quantities are presented infunction of the scalar vev in Figure 3.Figure 3 shows the relative difference between the sphaleron energy with themixing angle θ W = 0 and θ W = 0 and also the (negative) dipole energy of thesphaleron. It turns out that for any scalar representation, the relative differencebetween the sphaleron energy with θ W = 0 and θ W = 0 is always less than 1% and– 10 – -3 -2 -1
50 100 150 200 250 300 350 - ∆ E S ph / E S ph υ (GeV)(A) (1/2,1/2)(1,1)(3/2,1/2)(3/2,3/2)(2,1)(2,2) 10 -3 -2 -1
50 100 150 200 250 300 350 - ∆ E S ph / E S ph υ (GeV)(B) (1/2,1/2)(1,1)(3/2,1/2)(3/2,3/2)(2,1)(2,2)10 -3 -2 -1
50 100 150 200 250 300 350 - ∆ E S ph / E S ph υ (GeV)(C) (1/2,1/2)(1,1)(3/2,1/2)(3/2,3/2)(2,1)(2,2) Figure 3 . The relative difference in the sphaleron energy between the non-zero and zeromixing cases versus the scalar vev for different scalar representations, where the differenceis estimated: exactly (left), using the dipole approximation with U (1) gauge field effectneglected, Eq.(3.3) (right), and the case with U (1) gauge field effect considered, Eq.(3.5)(down) remains constant for different values of scalar vev. However, when considering the U (1) X gauge field effect on the dipole energy Eq.(3.5), it becomes closer to the exactdifference.Now we present the sphaleron energy Eq.(2.17) as a function of the scalar vevfor different scalar representations as shown in Figure 4.In Figure 4 we can see that the sphaleron energy depends on the scalar vev witha slope that depends on scalar isospin J and hypercharge X (or J ). This allows usthe write the scaling law as E sph ( v, J, X ) = Z ( J, X ) v, (3.6)where the function Z ( J, X ) represents the slope in Figure 4.– 11 – E S ph ( G e V ) υ (GeV) (1/2,1/2)(1,0)(1,1)(3/2,1/2)(3/2,3/2)(2,0)(2,1)(2,2) Figure 4 . The sphaleron energy versus the scalar vev for different scalar representations.
Before the electroweak phase transition
T > T c , the classical background scalar field, φ c , is zero and the Universe is in the symmetric phase. In this phase, the sphaleronprocesses are in full thermal equilibrium and are given as [39–42]Γ sym ∼ α w T ln(1 /α w ) , (4.1)with α w = g / π is the weak coupling. Therefore any generated baryon asymmetrydue to the sphaleron processes will be erased by the inverse process. Once thetemperature drops below the critical one T < T c , bubbles of true vacuum ( φ c = 0)start to nucleate where the rate is suppressed as Γ ∼ exp ( − E sph /T ).The sphaleron decoupling condition indicates that the rate of baryon number vi-olation must be much smaller than the the Hubble parameter [8, 9, 43] and therefore,the condition on the sphaleron rate is [10, 15, 44] − B dBdt ≃ N f π ω − α w κ N tr N rot e − E sph /T < H ( T ) , (4.2)where B is the baryon number density, the factors N tr and N rot come from the zeromode normalization, ω − is the eigenvalue of the negative mode [45]. The factor κ isthe functional determinant associated with fluctuations around the sphaleron [13].It has been estimated to be in the range: 10 − . κ . − [18, 46]. The Hubbleparameter is given as H ( T ) ≃ . p g ∗ ( T ) T /M pl , (4.3)where M pl and g ∗ are the Planck mass and the effective number of degrees of freedomthat are in thermal equilibrium. The term ”sphaleron processes” is used in the literature to refer to the baryon number violatingprocesses which also have the CP violating feature. – 12 –t was shown in [23] for the doublet case (
J, X ) = (1 / , /
2) that the sphaleronenergy at a given temperature can be well approximated by the following relation E sph ( v ( T ) , T ) v ( T ) = E sph ( v ) v , (4.4)where v ( T ) is the vev of the scalar field at temperature T and v is its zero tem-perature value. Eq.(4.4) shows that a straightforward estimation of the sphaleronenergy at finite temperature is possible by determining its energy at zero tempera-ture. This means that the scaling law Eq.(3.6) is valid also at finite temperature case,where the function Z ( J, X ) is temperature-independent. Because of similar linearscaling shown by higher scalar representations in Figure 4, we can use the scalinglaw Eq.(3.6) for other representations.Hence, for general scalar representation, the decoupling of baryon number viola-tion Eq.(4.2) implies the following relation [44] v ( T c ) T c > Z ( J, X ) h .
97 + ln( κ N tr N rot ) + ln ω − m W − ln g ∗ . − T c GeV i . (4.5)Most of the parameters in the r.h.s of Eq.(4.5) are logarithmically model-dependentand therefore one can safely use the SM values. In the case of SM, we have N tr N rot ≃ .
13 [10] and for λ/g = 1, ω − ≃ . m W [15, 18, 45]. It can be noted that thecontributions of model dependent quantities in v ( T ) /T are smaller than Z ( J, X ), forexample, in the SM [44] zero mode contribution is around 10% and the contributionsfrom the negative mode, relativistic degrees of freedom and critical temperature areabout 1%. For this reason we can consider the dominant contribution is coming from Z ( J, X ). In conjunction, using κ = 10 − (or 10 − ), g ∗ ≃ .
75 and T c ≃
100 GeV,we have from Eq.(4.5), v ( T c ) T c > η J,X , (4.6)where η J,X is given for each scalar representation in Table-1.It is clear that as the representation becomes larger, the strong first order phasetransition criterion gets relaxed. Generally, the case of κ = 10 − is the commonlyused criterion in the literature. In a general case of a multi-scalars model withrepresentations ( J ( i ) , X ( i ) ), the criterion Eq.(4.6) can be generalized asΘ( T c ) T c > , (4.7)with Θ( T c ) = X i v i ( T c ) η J ( i ) ,X ( i ) , (4.8)with v i ( T ) is the temperature dependent scalar vev of the multiplet Q ( i ) . In orderto check the criterion Eq.(4.8), we consider the case of a model with two scalar– 13 – X Z ( J, X ) η J,X ( κ = 10 − ) η J,X ( κ = 10 − )1/2 1/2 36.37 1 . . . . . . . . . . . . . . . . Table 1 . The values for the parameters Z ( J, X ) and η J,X for different scalar representa-tions. representations and estimate the ratio E sph / Θ for different values of J , J , X , X , v and v while keep the W gauge boson mass constant. The ratio E sph / Θ versusthe ratio v /v is shown in Figure 5.
30 35 40 45 50 0.1 1 10 E s ph / Θ υ / υ [(1/2,1/2),(1/2,1/2)][(1/2,1/2),(1,0)][(1/2,1/2),(1,1)][(1/2,1/2),(3/2,1/2)][(1/2,1/2),(2,1)][(1,0),(2,0)][(1,1),(3/2,3/2)][(3/2,1/2),(3/2,1/2)] Figure 5 . The sphaleron energy versus the scalar vev for different scalar representations.The self quartic couplings of scalar multiplet Q ( J , X ) ( Q ( J , X )) is set to 0 .
12 (0 . . From Figure 5, it is clear that the sphaleron energy scales like Θ for differentrepresentations and vevs within the error less than 5.7 %; and if the values of thetwo vevs are comparable, this error is reduced to 2.7 %. Therefore, one can safelyuse Eq.(4.8) as a criterion for a strong first order phase transition in any model withmultiscalars. – 14 –
Conclusion
We have constructed the energy functional and relevant variational equations ofthe sphaleron for general scalar representation charged under SU (2) × U (1) X gaugegroup and shown that the sphaleron energy increases with the size of the multiplet.Furthermore, it has been shown that at a fixed value of the vev, the sphaleronenergy is large for larger representation and for each representation, it linearly scaleswith the vev. As the energy of the sphaleron increases with the size of the scalarrepresentation, the criterion for the strong first order phase transition is relaxed forlarger representation. We have presented a representation dependent criterion forstrong phase transition which is relevant for the electroweak baryogenesis.We have also found that the dipole approximation (with or without considering a i in the U (1) X current, j i ) does not correspond exactly the energy difference E sph ( g ′ =0) − E sph ( g ′ = 0) and that is less than 2% for any scalar representation. In this casethe U (1) X field profile is just a deviation from unity and therefore just playing arelaxing role similar to singlet seen in [26].However, as we have seen in Figure 3 that the dipole contribution to the sphaleronenergy is negative, its coupling with the external magnetic field produced in thebubbles of first order phase transition through the dipole moment would lower thesphaleron energy and thus strengthen the sphaleron transition inside the bubble andwash out the baryon asymmetry more efficiently as pointed out in [47]. A morecareful analysis on this aspect for the sphaleron with higher scalar representationwill be carried out in [38].We have presented in Eq.(4.8) a general criterion for the strong first order phasetransition in a model with multiple scalars of different representations ( J, X ) and wehave shown that this approximate criterion is valid with an error less than 5%.
Acknowledgements
We are grateful to Goran Senjanovi´c, Eibun Senaha, Andrea De Simone, DietrichB¨odeker and Xiaoyong Chu for the critical reading of the manuscript and helpful com-ments. T.A.C. would also like to thank Basudeb Dasgupta, Luca Di Luzio and MarcoNardecchia for discussion. A.A. is supported by the Algerian Ministry of Higher Ed-ucation and Scientific Research under the CNEPRU Project No.
D01720130042 . A Asymptotic solutions
To capture the dependence of solutions on (
J, X ), in this section we have includedthe analytical estimates of solutions for the asymptotic region ξ → ξ → ∞ . Forthe energy functional Eq.(2.17) to be finite, the profile functions should be f ( ξ ) → ( ξ ) → f ( ξ ) → h ( ξ ) →
0. Therefore, at ξ ∼
0, the equations Eq.(2.19) arereduced into ξ f ′′ − f + 2 f + αξ h = 0 , (A.1) ξ f ′′ − f + 4 f + βξ h = 0 , (A.2) f ′′ + 2(1 − f ) − ( g ′ g ) βξ h = 0 , (A.3) ξ h ′′ + 2 ξh ′ − m h = 0 , (A.4)where m = Ω v (2 α + β ) . (A.5)The solution of Eq.(A.4) which leads to finite energy of the sphaleron is h ( ξ ) ∼ Aξ − (1 − p ) , (A.6)with p = r m. (A.7)Now at ξ ∼ f ( ξ ) ∼ f ( ξ ), so using this approximation, from Eq.(A.1) we have, f ( ξ ) ∼ Bξ − Aαξ (3+ p ) ( p − p + 5) . (A.8)On the other hand, we have considered f ( ξ ) as a perturbation in Eq.(A.2). Therefore,we have f ( ξ ) ∼ Cξ + Bξ − Kξ (3+ p ) . (A.9)Here, K is defined as follows K = 3 A { α (3 p − m + 3) + 8 mβ (4 m − } m (4 m − m + 3 p − . (A.10)Finally from Eq.(A.3), we have f ( ξ ) ∼ Dξ + 3 Aβg ′ ξ (3+ p ) g (3 p − m + 3) , (A.11)and A , B , C and D are integration constants.On the other hand, for asymptotic region, ξ ∼ ∞ , all the profile functionsmust approach unity to have finite energy of the sphaleron. So we consider thefunctions to be the small perturbation to unity as follows. Taking, f ( ξ ) = 1 + δf ( ξ ),– 16 – ( ξ ) = 1 + δf ( ξ ), f ( ξ ) = 1 + δf ( ξ ) and h ( ξ ) = 1 + δh ( ξ ) and keeping only thelinear terms of the variation, we have δf ′′ − αδf = 0 ,δf ′′ + β ( δf − δf ) = 0 ,δf ′′ − g ′ g β ( δf − δf ) = 0 ,ξ δh ′′ − ξδh − λv g Ω ξ δh = 0 . (A.12)The asymptotic solutions at ξ ∼ ∞ are, f ( ξ ) ∼ Ee −√ αξ ,f ( ξ ) ∼ F e −√ βξ ,f ( ξ ) ∼ Ge −√ βξ ,h ( ξ ) ∼ He − √ λvg Ω ξ ξ , (A.13)where E , F , G and H are again integration constants. The constants from A to H depend on ( J, X ) and couplings and they are determined by matching the corre-sponding asymptotic solutions and their first derivatives at ξ = 0. Therefore afterthe matching, the integration constants are, v and v H = − ( p − e v Ω n ( p + 1) + v Ω n , A = 1 + He − v Ω n ,E = − e √ α √ α + 2 (2 + 2 Aα (1 − p )( p − p + 5) ) , B = 1 + Ee −√ α + 4 Aα ( p − p + 5) ,B = 1 + Ee −√ α + 4 Aα ( p − p + 5) , F = e √ β √ β + 3 ( − B −
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