Sphaleron in the first-order electroweak phase transition with the dimension-six Higgs operator

By adding the dimension-six operator for the Higgs potential (denoted $\mathcal{O}_6$) in Standard Model, we have a first-order electroweak phase transition (EWPT) whose strength is larger than unity. The cutoff parameter of the dimension-six Higgs operator ($\Lambda$) is found to be in the range 593-860 GeV with the Wilson parameter equals to unity; it is also shown that the greater the $\Lambda$, the lower the phase transition strength and the larger the Wilson parameter, the wider the domain of $\Lambda$. At zero temperature, the sphaleron energy is calculated with a smooth ansatz and an ansatz with scale-free parameters, thereby we find that smooth profiles are not more accurate than profiles with scale-free parameters. Then, using the one-loop effective Higgs potential with the inclusion of $\mathcal{O}_6$ instead of all possible dimension-six operators, we directly calculate the electroweak sphaleron energy at finite temperature with the scale-free parameters ansatz and show that the decoupling condition is satisfied during the phase transition. Moreover, we can reevaluate the upper bound of the cutoff scale inferred from the first-order phase transition. In addition, with the upper bound of the cutoff parameter (about 800-860 GeV), EWPT is a solution to the energy scale of the six dimension operators, there is an extended conclusion that EWPT can only solve at a large energy scale than that of SM.

theory valid up to a certain scale of energy called the cut-off scale (or the new-physics scale); this class of theory is called SMEFT (Standard Model Effective Field Theory). Of course, we expect that the results in SMEFT should reduce to those in SM in the limit that the cut-off scale goes to infinity, which is equivalent to the assumption that SM is valid at arbitrary high energy. Though, some care is required in making this statement since the Planck scale is believed to be the scale of energy where quantum gravity effects come into play.
SMEFT includes higher-dimensional, nonrenormalizable operators such as dimensionfive or dimension-six operators with a number of the so-called Wilson coefficients. There is nothing to prevent us from adding higher-dimensional operators, but there are also no solid physical reasons for adding them. The inclusion of these operators is just a phenomenological approach to achieve desired results; it is not originated from, for example, the defects in the theoretical framework of SM or a realization of a more reliable theory. Now, we will focus on the aspects of SMEFT related to EWPT; and for this aim, the dimension-six operators are simplest, good choices since, as we will see, they will affect some properties of EWPT in the way that we want. There are totally 20 possible dimension-six operators in the electroweak sector, but we want to consider only the dimension-six Higgs operator for the Higgs potential, denoted O 6 . There are some technical and physical reasons for this choice. Firstly, if we consider different dimension-six operators at a time, we will not be able to isolate the value of the cut-off scale Λ because there are many different Wilson coefficients. In other words, if we only consider O 6 , we can absorb the coefficient in front into the cut-off scale and hence calculations are simplified and definite statements about Λ could be made. Secondly, the O 6 operator is the only dimension-six operator that can affect the form of the effective potential and can shift the strength of EWPT, which is connected to the third condition of Sakharov. Unfortunately, this operator does not have contributions to C and CP violations and therefore, we say that in the context of SMEFT the inclusion of O 6 is required but not satisfactory to completely solve the baryogenesis problem. However, as we will see, the cut-off scale Λ is strongly constrained in order to have sufficient phase transition strength. Due to the fact that no new-physics phenomena are detected at the TeV scale, SMEFT with dimension-six operators is strongly suppressed and is not a prominent candidate to tackle the baryogenesis problem. Further inclusions of dimension-six operators that involve C and CP violations will not help unless the Wilson coefficient of O 6 is unreasonably large compared to the values obtained from standard fitting methods [1].
Another interesting finding is that EWPT is independent of the gauge. So the EWPT only needs to be calculated in the Landau gauge, which is sufficient and also physically suitable [19,22,23]. The damping effect in the thermal self-energy term or daisy loops have small distributions [19]. It was approximated g 2 T 2 /m 2 (g is the coupling constant of SU(2), m is mass of boson), m ∼ 100 GeV, g ∼ 10 −1 so g 2 /m 2 ∼ 10 −6 . Therefore, this distribution does not change the strength of EWPT or, in other words, it is not the nature of EWPT.
There are some recent notable results with the O 6 operator in the EWPT and sphaleron problems. The prediction about a first-order phase transition when there was an inclusion of O 6 with different values of Higgs mass and cut-off parameters was established in Ref. [24]. Refs. [25,26] also made a prediction of the first-order phase transition and estimated sphaleron energy at zero temperature, although the contribution of particles has not been calculated in details yet. The tree-level EWPT sphaleron energy was solved by numerical methods at zero temperature with the dimension-six operators in Ref. [27]. Accordingly, in this paper, we attempt to constrain the range of the cut-off scale. While the lower bound the cut-off scale is determined by some mathematical arguments of the treelevel Higgs potential at zero temperature, the upper bound of the cut-off scale is determined from the requirement of EWPT's strength and is reassessed with the sphaleron energy at finite temperature using the scale-free parameters ansatz. In addition, with operators O 6 associated with the scenario of cosmic inflation, Kusenko came up with a chart to calculate the baryon asymmetry of the Universe (BAU) at about 10 −10 [28].
There is one point we should clarify: the concept of a first-order EWPT should not be thought of as equivalent to the concept of departure from thermal equilibrium, although a first-order EWPT may indeed lead to the departure from thermal equilibrium. A first-order phase transition is defined as a kind of phase transition that happens when the two minima are separated by a potential barrier; this kind of phase transition is violent and the symmetry is broken via bubble nucleation. The departure from thermal equilibrium is defined more rigorously in the context of topological phase transition as Γsph ∼ exp where Γsph and E sph are the sphaleron rate and the sphaleron energy, respectively. In SM, this condition is translated into v c /T c > 1 by using the approximate scaling relation ; however, in beyond SM models this scaling relation may break down. The interesting point of this paper is to investigate both the EWPT and the sphaleron solutions simultaneously so that we can understand the third condition of Sakharov better and, as we will see, we can assess the upper bound of the cut-off scale more thoroughly.
We have reviewed the current core results. Based on that, we will calculate EWPT and sphaleron in SM with O 6 operator in this article, which is organized as follows. In section II, we summarize some main features of SMEFT. In section III, we derive the effective potential which has the contribution from the O 6 operator. We analyze in details the phase transition and find that the phase transition is first-order and show constraints on the cut-off scale.
In section IV, electroweak sphalerons are calculated at zero and high temperature using the effective potential with the O 6 operator. Finally, we summarize and make outlooks in section V.

II. SMEFT
The main idea of SMEFT is that SM is just an effective theory at low energy. In other words, SM is essentially a leading order approximation of a more fundamental theory in the expansion of EFT with Lagrangian as follows where L SM is the Lagrangian of SM, Λ is the cut-off scale (or new physics scale), the expansion factors C k are Wilson parameters and Q k (s) are higher dimensional operators.
Because the couplings have negative energy dimensions for operators of dimension-five or more, SMEFT is not renormalizable.
There are three conditions that SMEFT must satisfy [29]: Firstly, the high-order operators must satisfy the gauge symmetries in SM, which is  [35]. In this article, we will stop at 1-loop level.
The one loop Higgs effective potential when considering the contribution of heavy particles, which are usually drived in two ways. The first is functional approach, the second is the perturbation one. The high temperature potential is usually derived by using the Bose-Einstein or Fermi-Dirac distributions or using the finite-temperature Green function.

The functional approach
The first method was used by Coleman and Weinberg in 1973 [33], when using functional integrals and the finite temperature Matsubara Green function to derive an one loop effective potential which has the contribution of all particles. The loops are fermion, gauge and neutral bosons loops and the external lines of these loops are the Higgs fields.
The action is given by The generating functional Z[j] is the transition amplitude from the vacuum in the far past to the one in the far future, with the source j, where The connected generating functional W [j] is defined by The effective action Γ[φ] is the Legendre transform of W [j] as follows:: with From Eqs. (6), (7) and (4), it is easy to prove that The above equation shows that there is no external source (j), φ is the vacuum.
The Taylor expansion of connected generating and generating functional for j, yields The Green's functions G (n) are the sum of all Feynman diagrams with n external lines.
Whereas, the expansion coefficients of iW [j] are connected Green's functions; G c (n) is the sum of all Feynman diagrams associated with n external lines. Similarly, we have a Taylor expansion of the effective action in the φ(x) field as follows: Γ (n) is the sum of all one-particle-irreducible (1PI) Feynman diagrams with n external lines. The Fourier transform of the effective action in φ(x) is From the two above Fourier transforms, the effective action in (10) is rewritten as follows The Fourier transform of φ c is The effective potential becomes, Meanwhile, the definition of the effective potential is Comparing the above two equations (14) and (15), we get The Feynman rules at zero and finite temperature are shown in the table I [36], which is used to calculate the Green functions Γ n or the sum of all diagrams.
The existence of dimension-six operators will affect many scattering and decay processes happening in particle accelerators, so the nonzero Wilson coefficients inferred from fitting method will be the indication of new physics. However, most of the Wilson coefficients except c 6 are tightly constrained around zero [1]. The difficulty in constraining c 6 lies in the fact that the Higgs potential in the SM is conventionally assumed to be of the form ∼ φ 2 + φ 4 , but the Higgs self-coupling λ is poorly measured. Moreover, with our choice of the form of O 6 it will not affect λ anyway; in other words, new physics cannot be inferred from the measurement of λ but has important effects on EWPT. So it is reasonable to work solely with O 6 in this paper.
We also note that this 1-loop effective potential has already included some dimensionsix operators other than O 6 . For example, the 1-loop gauge boson term corresponding to O r ∼ W 2 φ 4 is included in the Γ (n) function with n = 4 external lines of the Higgs field. The which cannot be implemented in the calculation of the effective potential but it does have contribution to the sphaleron energy; we will not consider it in this paper.
The next work is from Lagrangian of each field, we will calculate all their contributions by summing all the Green functions in Eq. (16) and renormalization. We will have two components, one is quantum contributions, the other is thermal contributions. The thermal contribution is obtained by the finite temperature Matsubara Green function (or temperature propagators in the table I).

The perturbation approach
Firstly, we consider a toy model describing a self-interacting real scalar field and we see that this scalar field can be Higgs field. The scalar field satisfies the following equation of motion [37] where V ′ = ∂V ∂χ , potential V is unknown. The field χ can always be decomposed into homogeneous and inhomogeneous components by adding a contribution from thermal fluctuation φ: Using Taylor series for V (χ + φ), we have Noting that ∂φ ∂χ = 1 and Substituting (18) into (17), and we only consider the potential component and averaging over space, we obtain The two last components in (19) can be rewritten as the derivative of an effective potential. In scalar field theory, we can get Taking into account that m 2 φ = V ′′ . We can rewrite the third term in (19) as Using Taylor expansion for (21), we obtain in which The occupation numbers n k are given by the Bose-Einstein formula, Substituting ν = 1, we obtain an expansion of J 1 Therefore we obtain and with In case α = m φ T , we have Substituting Eq. (30) in Eq. (29) yields So Eq. (31) is equivalent to Eq. (28).
V ∞ can be absorbed by a redefinition of constant in the original potential. The potential V is The final result, which includes both quantum and thermal contributions, is and at zero temperature the last term in (33) will vanish. Then the effective potential reduces to Therefore, if we calculate the contributions of other particles and perform the renormalization process, we get the full 1-loop effective potential.
In addition, we have another way when the involvement of the gauge and ring loops, by using functional methods, Nielsen identities and the small expand as in articles [18,19].
However, we also see that the gauge and ring loops do not play a major in the phase transition problem.
Ref. [30] has shown the contributions of the dimension-six operators in the energy of sphaleron at one loop level. In the calculation of the one loop effective Higgs potential as mentioned in the functional approach, we have summed up all the contributions of one-loop diagrams. So in the effective potential, there are always the contributions of some dimensionsix operators. Apart from daisy loops, but we know that the thermal contributions of daisy loops are quite small, they can be ignored, this was also mentioned in the introduction section. Finally, when calculating the effective potential to investigate the EWPT and the sphaleron, we only need to use the O 6 operator, regardless of the other dimension-six operators.
B. Effective potential at T = 0 The O 6 operator is an addition to the Higgs potential, with all the contributions of SM bosons and top quark, Eq. (16) becomes Selecting the O 6 operator for the Higgs scalars so that it does not change the parameters and mass of the Higgs bosons. Specifically, The normalization conditions for such the potential are: From these conditions, expanding the logarithmic functions, we get The vacuum contribution of the effective potential is depicted in the figure 1 with Λ = 800 GeV. The vacuum contribution (T = 0) of SMEFT increases more rapidly and is larger than that of the SM in the range φ > 246 GeV. In contrast, the behavior of the potentials in the range φ < 246 GeV is opposite to that in the range φ > 246 GeV. This suggests that the minimum of SMEFT is more stable than that of the SM. where Using Eq. (37), we obtain the explicit expressions of the normalization parameters which was derived from the 1-loop effective potential at zero temperature, as follows Here, we have put The transition-ending temperature is the temperature at which the initial minimum becomes a maximum: The above expansion for a high-temperature effective potential will be better than 5% if m boson T < 2.2 [38], where m boson is the relevant boson mass. The mass range of SM particle in the EWPT calculations always fit with this because the critical temperature is about 100 GeV.
The effective potential is drawn as shown in the figure 2. It appears the non-zero second minimum and the value of this minimum grows as temperature decreases. Between two minima, a barrier appears which is a sign of first order phase transition.
The results of phase transition in the range 700 GeV < Λ < 1000 GeV, are summarized in the table II. An interesting conclusion is that when Λ increase, T 1 (the second minimum starts to appear from here) and the critical temperature (T c ) approached each other. As where Ω is a cosmological constant and λ > 0. Here we have temporarily written down explicitly the Wilson coefficient of O 6 since we will also have the chance to discuss its sign.
We find the local extremes of potential by calculating its first derivative:  here we have set A ≡ 3c 6 4Λ 2 for brevity, and the discriminant ∆ of the quartic equation is On the other hand, since O 6 does not affect three normalized conditions: h so the coefficients in the Higgs potential are still the same form as those of SM: Substituting µ 2 = λv 2 into Eq.(41), we get The solutions of the quartic equation are At zero temperature, we expect that Eq.(40) should have three solutions corresponding to the three extremes at φ = 0 and φ = ±v, so we must have the following condition: We see that the case c 6 < 0 must be removed. With c 6 = 1 and substituting the experimental values v = 246 GeV and m h = 125 GeV into (43), we get Λ > 593 GeV.
Therefore the larger the Wilson parameter, the bigger lower bound, so overall we have:

IV. THE ENERGY OF SPHALERON
In a non-Abelian gauge theory such as the electroweak theory there is another interesting kind of transition called topological phase transition, which is the transition of the field configurations between topologically distinct vacua. When such a transition occurs, the baryon number will be violated and it opens up the possibility for the electroweak theory to satisfy the first condition of Sakharov. At zero temperature, the instanton process is strongly suppressed due to the smallness of the weak coupling [37]. On the other hand, at high temperature the sphaleron process dominates and the transition rate is proportional to the Boltzmann factor as Γsph ∼ exp(−E sph /T ). When the temperature is high, the "height" of the potential, which is the energy of sphaleron, is small and thermal equilibrium can be obtained. However, when the temperature is low, the height of the potential barrier will rise and the sphaleron rate must be strongly suppressed. The universe also expands while this phase transition occurs, so the condition for the departure from thermal equilibrium is According to Ref. [27], the sphaleron energy at zero temperature at tree level is calculated by including all relevant dimension-six operators. However, the first difficulty is that it is unclear whether the Wilson parameters are positive or negative; the second difficulty is that the equations of motion are temperature-dependent when considering EWPT and hence are even more difficult to be solved.
In addition, the dimension-six operators are not renormalizable; however, their contributions must be actually smaller than the contribution of the tree-level or theoretically new physics will not be true in nature. Therefore, when all dimension-six operators are taken into account, there will be a greater contribution than the tree level, which is difficult to handle. Moreover, the bigger the electroweak sphaleron energy, the smaller the B violation; this energy should only be larger than what satisfies the decoupling condition (we will see this in the following subsections).
Finally, we calculate this high temperature sphaleron energy by using the high temperature effective potential in the previous section. Because according to the effective potential calculation, it has always entered some dimension-six operators at the 1-loop level (with the external lines are the Higgs fields) and the renormalization of the effective potential is also implicitly understood to renormalize those operators.

A. Sphaleron ansatz
Manton and Klinkhamer [39] have found the ansatz approaches to find an approximate sphaleron solution without fully solving all the equations of motion. The contribution to the sphaleron energy of the hypercharge gauge field is found to be very small, so we will not consider them (i.e. B µ = 0) [39]. We also choose the temporal gauge, so W a 0 = 0. Let n i ≡ x i r where r is the radial coordinate in the spherical coordinates and assume that sphaleron has a spherically symmetric form. The sphaleron ansatzes then have the following with the boundary conditions are The functions f (r) and h(r) are called radial functions (or profile functions). The first boundary condition is intended to avoid singularity at r = 0, and the second boundary condition is to ensure that the ansatzes are asymptotic to the form of the fields at infinity [39].
As we will see, in this way the equations of motion will be converted to the equations of h(r) and f (r) and everything will be much simpler.
Next, in order to calculate the sphaleron energy, we need to solve all the equations of motion to find h(r) and f (r) and then substituting these functions into the spahleron energy function. However, solving these equations of motion is also very complicated, can only solve by numerical method as in Ref. [27]. To be more streamlined we can assume the above profiles by spherical functions, and refine these functions through free parameters.
The calculation steps are shown in the next section.

B. Contributions to the sphaleron energy
The sphaleron energy functional takes the form • The first term in Eq.(47) is the contribution of the isospin gauge fields 1 4 W a kl W a,kl : Substituting the ansatzes in (45) into the first term in (47) yields So the sphaleron energy contribution of the gauge fields is We can use the following dimensionless distance variable So we get the sphaleron energy of the gauge fields in terms of the variable ξ as follows • The second term in Eq.(47) is the kinematic of the Higgs field (D k H) † (D k H).
Following the same procedure as we did in the previous case, the contribution of the kinematic term of the Higgs field to the sphaleron energy is (50) • The third term in Eq.(47) is the Higgs potential with O 6 at tree-level.
V Higgs (H, The difference between sphaleron energy at zero temperature and sphaleron energy at finite temperature lies solely in the Higgs potential. The form of contributions from kinematic terms remains the same, but all of the profile functions and the dimensionless distance must be rescaled when calculating sphaleron energy at nonzero temperature.

C. Profile functions
There are two kinds of ansatz: the ansatz with scale-free parameters firstly introduced in [39], and the smooth ansatz motivated from kink-type solutions [27,40]. We will use both of these two types to calculate the sphaleron energy at zero temperature and compare it to the values obtained from numerical calculations, then we will use the better ansatz to calculate sphaleron energy at finite temperature.
For the further presentation, we use the following ansatz 1. Ansatz a: f and h with scale-free parameters with a, b are scale-free parameters.
2. Ansatz b: f and h are smooth functions where A and B are free parameters.
Note that we must select the profile functions that satisfy the boundary conditions in (46). They do not necessarily satisfy the nonlinear equations (52) and (53). However, as we shall see, finding an ansatz satisfying the equations of motion at a certain asymptotic limit, especially in the sphaleron core region, will give better results.
D. Sphaleron energy at zero temperature at tree level with O 6 The zero temperature sphaleron energy at tree-level with O 6 is As mentioned in the introduction, the Wilson coefficient of O 6 can be absorbed into Λ.
Assuming that the Higgs potential in the SM has the standard form ∼ φ 2 + φ 4 , we have λ ≈ 0.1. Because choosing the O 6 operator does not change the Higgs mass form, so we still have the value of λ like in the SM.
1. Ansatz a (Eqs. (52) and (53)) With ansatz a, Eq.(55) becomes: To minimize this energy according to the parameters a and b, we get the energy value   With ansatz a, the sphaleron energy in the SM (Λ = ∞) is about 2.024 × 4πv g (TeV). Comparing this value to the value obtained from numerical method in Ref. [27], we find that  ansatz a is in good agreement and the difference is only about 5.6 %, while ansatz b gives a difference of about 11.9%. In addition, the sphaleron energy which has the contribution of O 6 at zero temperature, with Λ ranging from 590 GeV to 860 GeV, is in the range [27]. So according to the tables III and IV, ansatz a also gives better results. Therefore, we will only use the ansatz a to calculate the sphaleron energy at nonzero temperature at one-loop level.

E. Sphaleron energy at finite temperature at 1-loop level with O 6
Substituting the 1-loop effective potential (Eq. (38)) into the sphaleron energy functional, we obtain the sphaleron energy at the temperature T : In the electroweak phase transition in the section III, the effective potential depends on φ. In the sphaleron problem, we want to express it through the profile functions h, f . From the ansatz of the Higgs doublet and gauge field in the equation (45), we obtain On the other hand, ignoring the contribution of the Goldstone bosons, the Higgs doublet is so we have the following relationship φ = vh.
The above equation allows us to convert the effective potential in φ to that in h.
When calculating the sphaleron energy at temperature T , to be comparatively consistent with the sphaleron energy at zero temperature, ansatz a must be used. However, it should be noted that we must rescale the variable ξ and the radius function h(ξ) as follows: To avoid cumbersome, we will ignore the tilde symbol and automatically perform the above operation. The effect potential is given by In the above equation, in the first line, we only rewrite the effective potential in the section III; in the second line, we replace φ = vh and the radius function h; in the third line, we add and remove the temperature-dependent constants and set it to C(T ). The sphaleron energy will be rewritten The constant C(T ) is removed so that the sphaleron energy does not diverge. This is completely normal because this cosmological constant can be absorbed into the three normalized conditions of the effective potential. The result is given in the table V. We see that the sphaleron energy decreases with increasing temperature. This is consistent with our expectation of the large baryon violation rate in the early universe. For comparison purposes, we showed in the last column the sphaleron energy calculated from the well-known scaling expression [41]: In [27], the authors only used the above approximate expression and did not calculate directly the sphaleron energy at nonzero temperature from the effective potential.
The profile functions of sphaleron for the Higgs boson and isospin gauge fields are plotted in the figures 5 and 6, respectively.  The sphaleron solutions are only physically meaningful at T ≤ T c , although these solutions can be constructed mathematically up to T 1 . Unlike the bubble solution, sphaleron solution still exist at zero temperature and hence baryon number violation process is absent.
The question of whether or not we can recreate the conditions of the early universe, and hence detect a process that violates baryon number in particle accelerators, is very deep and remains unanswered.
In order to ensure that the baryon number is preserved during the expansion of the universe, we need the baryon washout avoidance condition, which is also known as the decoupling condition or the third condition of Sakharov. The baryon-number violation rate must be smaller than the Hubble rate during the phase transition process (i.e. when the temperature drops from T c to T 0 ). On the other hand, EWPT occurs after inflation and belongs to the radiation-dominated period so that the Hubble rate is H 2 rad = 4π 3 g * 45m 2 pl T 4 . The authors in Ref. [27] carefully calculated the baryon number violation rate and gave results from the condition Γsph ≪ H rad as: In summary, by calculating the sphaleron energy at a finite temperature directly from the effective potential, we can give a more thorough assessment of the thermal imbalance condition, which is commonly known through v c /T c > 1. This condition is only inherently reliable in the SM. In the previous section, we saw that the upper bound of the cut-off scale is 860 GeV to satisfy v c /T c > 1. By assessing the decoupling condition when calculating sphaleron energy we see that this upper bound is correct.  Green zone: the region that satisfies the decoupling condition.

V. CONCLUSION AND DISCUSSION
With the same calculation as the section IV, we can calculate the sphaleron energy with the contributions from other dimension-six operators as in Ref. [27]. We also estimate with our calculation has an error not exceeding 10% compared to the result calculated in Ref. [27].
These results are detailed and also consistent with the conclusions in the 2004 paper [24] that explored the EWPT problem in the SM with the O 6 operator at 1-loop. In this article, the exactly value of the Higgs mass is used to find the upper bound of Λ (about 860 GeV) that satisfies the out-of-thermal equilibrium condition. It also shows that the narrow temperature range where phase transition occurs. Among many possible dimension-six operators, we only consider the O 6 operator for the Higgs potential. This is not a sufficient condition but a prerequisite for explaining matter-antimatter asymmetry in the universe. Specifically, when the EWPT survey had O 6 , we found that the phase transition strength (v c /T c ) is large enough to cause thermal nonequilibrium. In addition, when investigating the sphaleron solution, the baryon number violations are preserved during phase transition. Thus, the first and third conditions of Sakharov are feasible considering SM + O 6 . To fully solve the problem of matter-antimatter asymmetry, we need to cleverly combine the O 6 operator with other dimension-six operators that cause C and CP violations. That is our possible future work.
We do not know the contributions of these sixth-order operators is positive or negative, so we cannot conclude whether the sphaleron energy when calculating the addition of sixthorder operators strongly satisfies the decoupling condition. However, in order to have a first order phase transition, in our calculation results the contribution from the dimension-six Higgs operator must be positive. This opens up the need to study Wilson's parameters more.
With the numerical result in [27], we find that, in the value domain (44), the contribution of O 6 operator does not exceed 5% in sphaleron energy when compared to the sphaleron energy of the SM at zero temperature. If other dimension-six operators are also constrained in this domain and all of them has positively contributions, the total contribution will be about 35%, this is a very significant contribution that will do the sphaleron energy increases greatly and may violate the decoupling condition. In the event that these contributions can compensate for each other, the sphaleron energy can only slightly increase compared to the SM. In a nutshell, there is a need for more information about the dimension-six operators, considering them in other quantum corrections, in the decay channels, which will allow us to calculate the sphaleron energy in more detail. In contrast, our value domain of the cut-off parameter and the EWPT and sphaleron problem are the good binding channels for finding new physics when adding dimension-six operators. These are also our upcoming calculations as well as checking these solutions with the Cosmotransition package [42].
These results are a continuation of the previous articles, show a comprehensive view and more complete results for the cut-off parameter as well as the temperature electroweak sphaleron in the appropriate ansazts. It helps to show that the dimension-six operators can be one of the good improvements.