PHYSICAL REVIEW D 101, 116010 (2020)

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

† ‡ Vo Quoc Phong ,1,2, Phan Hong Khiem ,1,2, Ngo Phuc Duc Loc,1,2,§ and Hoang Ngoc Long 3,4,* 1Department of Theoretical Physics, University of Science, Ho Chi Minh City 700000, Vietnam 2Vietnam National University, Ho Chi Minh City 700000, Vietnam 3Theoretical Particle Physics and Cosmology Research Group, Advanced Institute for Materials Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 4Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam

(Received 30 March 2020; accepted 1 June 2020; published 15 June 2020)

By adding the dimension-six operator for the Higgs potential (denoted O6) 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 (Λ) is found to be in the range 593–860 GeV with the Wilson parameter equal to unity; it is also shown that the greater the Λ, the lower the phase transition strength and the larger the Wilson parameter, the wider the domain of Λ. 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 O6 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 dimension-six operators. There is an extended conclusion that EWPT can only be solved at a large energy scale than that of SM.

DOI: 10.1103/PhysRevD.101.116010

I. INTRODUCTION The interesting thing is that gauge symmetry inevitably leads to the existence of gauge bosons which mediate the The Standard Model (SM) of particle physics has interactions between matter (fermions). Furthermore, the established many good results that agree with experiments Higgs mechanism of SM, which is basically changes of and gave us a clear framework of how matter interacts with variables, can generate masses spontaneously without each other. The cornerstone of SM is the concept of introducing by hands the mass terms of gauge bosons that symmetries such as Lorentz symmetry and gauge sym- might spoil the renormalizability of the theory. Starting metry. We want the theory to be consistent with special from a few simple ideas, we got exactly what we want: a relativity, which is the theory of space and time in the absence of gravitation, so we need a kind of external renormalizable theory that can describe matter and inter- symmetry called Lorentz symmetry that transforms the actions (except gravity) in the flat background spacetime. coordinates of space and time. Motivated from this, we also However, there are some fundamental problems that want to have a kind of internal symmetry called gauge have not been solved by SM and one of them is the matter- symmetry that transforms the dynamical fields themselves. antimatter asymmetry puzzle. If we want a dynamical explanation for this rather than a conjecture of initial conditions, we have to investigate electroweak phase *Corresponding author. transition (EWPT) process that happens in the early [email protected] † universe. In 1967, Sakharov proposed three conditions [email protected] ‡ [email protected] that a theory must have in order to solve the baryogenesis §[email protected] problem, which are baryon number violation, C and CP violation, and out-of-equilibrium condition. Nevertheless, Published by the American Physical Society under the terms of EWPT within the context of SM does not offer good the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to solutions since it does not satisfy sufficiently the last two the author(s) and the published article’s title, journal citation, conditions of Sakharov. Therefore, theories beyond SM and DOI. Funded by SCOAP3. have to be taken into account. A common choice is

2470-0010=2020=101(11)=116010(17) 116010-1 Published by the American Physical Society PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020) considering that SM is only an effective theory valid up to a EWPT problem will be one of the urgent issues. During certain scale of energy called the cutoff scale (or the new- the period from 1967 to the present, the EWPT has been physics scale); this class of theory is called SMEFT calculated in SM as in Refs. [2–4] and in theories beyond (Standard Model effective field theory). Of course, we SM or in many other contexts as in Refs. [5–25]. A familiar expect that the results in SMEFT should reduce to those in negative result is that the EWPT’s strength is only larger SM in the limit that the cutoff scale goes to infinity, which than 1 at a few hundred GeV scale with the mass of Higgs is equivalent to the assumption that SM is valid at arbitrary boson must be less than 125 GeV [2–4]. So far, the origin of high energy. Though, some care is required in making this a first-order EWPT can be heavy bosons or dark matter statement since the Planck scale is believed to be the scale candidates [5,7–11,16–18,26–28]. of energy where quantum gravity effects come into play. Another interesting finding is that EWPT is independent SMEFT includes higher-dimensional, nonrenormaliz- of the gauge. So, EWPT only needs to be calculated in the able operators such as dimension-five or dimension-six Landau gauge, which is sufficient and also physically operators with a number of the so-called Wilson coeffi- suitable [19,28,29]. The damping effect in the thermal cients. There is nothing to prevent us from adding higher- self-energy term or daisy loops have small distributions. dimensional operators, but there are also no solid physical The daisy loops are hard thermal loops, each of which reasons for adding them. The inclusion of these operators is contributes a factor of g2T2=m2 [19] (g is the coupling just a phenomenological approach to achieve desired constant of SUð2Þ, m is mass of boson), m ∼ 100 GeV, results; it is not originated from, for example, the defects T ∼ 100 GeV, g ∼ 10−1 so g2T2=m2 ∼ 10−2. When con- in the theoretical framework of SM or a realization of a sidering daisy loops, the improvement of effective potential more reliable theory. leads to a reduction of the strength of EWPT roughly by a Now, we will focus on the aspects of SMEFT related to factor 2=3 [30]. Therefore, this distribution does not make a EWPT; and for this aim, the dimension-six operators are big change to the strength of EWPT or, in other words, it is simplest, good choices since, as we will see, they will affect not the nature of EWPT. some properties of EWPT in the way that we want. There There are some recent notable results with the O6 are totally 20 possible dimension-six operators in the operator in the EWPT and sphaleron problems. The electroweak sector, but we want to consider only the prediction about a first-order phase transition when there dimension-six Higgs operator for the Higgs potential, was an inclusion of O6 with different values of Higgs mass denoted O6. There are some technical and physical reasons and cutoff parameters was established in Ref. [31]. for this choice. First, if we consider different dimension-six References [32,33] also made a prediction of the first- operators at a time, we will not be able to isolate the value order phase transition and estimated sphaleron energy at of the cutoff scale Λ because there are many different zero temperature, although the contribution of particles has Wilson coefficients. In other words, if we only consider O6, not been calculated in detail yet. The tree-level EWPT we can absorb the coefficient in front into the cutoff scale sphaleron energy was solved by numerical methods at zero and hence calculations are simplified and definite state- temperature with the dimension-six operators in Ref. [34]. ments about Λ could be made. Second, the O6 operator is Accordingly, in this paper, we attempt to constrain the the only dimension-six operator that can affect the form of range of the cutoff scale. While the lower bound the cutoff the effective potential and can shift the strength of EWPT, scale is determined by some mathematical arguments of the which is connected to the third condition of Sakharov. tree-level Higgs potential at zero temperature, the upper Unfortunately, this operator does not have contributions to bound of the cutoff scale is determined from the require- C and CP violations and therefore, we say that in the ment of EWPT’s strength and is reassessed with the context of SMEFT the inclusion of O6 is required but not sphaleron energy at finite temperature using the scale-free satisfactory to completely solve the baryogenesis problem. parameters Ansatz. In addition, with operators O6 associ- However, as we will see, the cutoff scale Λ is strongly ated with the scenario of cosmic inflation, Kusenko et al. constrained in order to have sufficient phase transition came up with a chart to calculate the baryon asymmetry of strength. Due to the fact that no new-physics phenomena the Universe at about 10−10 [35]. are detected at the TeV scale, SMEFT with dimension-six There is one point we should clarify: the concept of a first- operators is strongly suppressed and is not a prominent order EWPT should not be thought of as equivalent to the candidate to tackle the baryogenesis problem. Further concept of departure from thermal equilibrium, although a inclusions of dimension-six operators that involve C and first-order EWPT may indeed lead to the departure from CP violations will not help unless the Wilson coefficient of thermal equilibrium. A first-order phase transition is defined O6 is unreasonably large compared to the values obtained as a kind of phase transition that happens when the two from standard fitting methods [1]. minima are separated by a potential barrier; this kind of In addition, since Higgs boson was discovered at the phase transition is violent and the symmetry is broken via LHC, particle physics almost completed its mission that bubble nucleation. The departure from thermal equilibrium provides a more accurate understanding of mass. The is defined more rigorously in the context of topological

116010-2 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

− Γ ∼ … Esph ≪ phase transition as sph OðT; Þ expð Þ Hrad, III. ELECTROWEAK PHASE TRANSITION Γ T where sph and Esph are the sphaleron rate and the sphaleron IN SM WITH O6 OPERATOR energy, respectively, H is the Hubble expansion rate in the rad A. Summary of calculating effective Higgs potential radiation-dominated era, and O(T,...) is a preexponential factor which is very hard to calculate. In the SM, this The effective potential for quantum field theory was first condition is translated into vc=Tc > 1 by using the approxi- introduced by Heisenberg and Euler [38]. It was then ≈ 0 mate scaling relation EsphðTÞ ½vðTÞ=vEsphðT ¼ Þ;how- applied to the spontaneous symmetry breaking survey by ever, in beyond SM models, this scaling relation may break Goldstone et al. [39]. The one-loop effective potential was down. The interesting point of this paper is to investigate calculated by Coleman and Weinberg in 1973 [40]. For the both the EWPT and the sphaleron solutions simultaneously case of multiple loops, one can refer to the calculations of so that we can understand the third condition of Sakharov Jackiw in 1974 [41] and Iliopoulos et al. in 1975 [42].In better and, as we will see, we can assess the upper bound of this paper, we will stop at one-loop level. the cutoff scale more thoroughly. The one-loop Higgs effective potential when consid- We have reviewed the current core results. Based on that, ering the contribution of heavy particles is usually wewillcalculate EWPTandsphaleron in SM with O6 operator derived in two ways. The first one is the functional approach, in this paper, which is organized as follows. In Sec. II,we and the second one is the perturbation approach. The summarize some main features of SMEFT. In Sec. III,we high-temperature potential is usually derived by using derive the effective potential which has the contribution from the Bose-Einstein or Fermi-Dirac distributions or using the O6 operator. We analyze in detail the phase transition and the finite-temperature Green’s function. find that the phase transition is first order and show constraints on the cutoff scale. In Sec. IV, electroweak sphalerons are 1. The functional approach calculated at zero and high temperature using the effective O The first method was used by Coleman and Weinberg in potential with the 6 operator. Finally, we summarize and 1973 [40], when using functional integrals and the finite make outlooks in Sec. V. temperature Matsubara Green’s function to derive the one- loop effective potential which has the contribution of all II. SMEFT particles. The loops are fermion, gauge, and neutral bosons The main idea of SMEFT is that SM is just an effective loops and the external lines of these loops are the Higgs theory at low energy. In other words, the SM is essentially a fields. leading order approximation of a more fundamental theory The action is given by in the expansion of EFT with Lagrangian as follows: Z X X S ϕ d4xL ϕ x : 2 1 5 5 1 6 6 ½ ¼ f ð Þg ð Þ L L Cð ÞQð Þ Cð ÞQð Þ SMEFT ¼ SM þ Λ k k þ Λ2 k k k k The generating functional Z½j is the transition amplitude 1 O from the vacuum in the far past to the one in the far future, þ 3 ; ð1Þ Λ with the source j, Z L Λ where SM is the Lagrangian of SM, is the cutoff scale Z½j¼h0 j0 i ≡ Dϕ expfiðS½ϕþϕjÞg; ð3Þ (or new physics scale), the expansion factors Ck are Wilson out in j parameters, and QkðsÞ are higher-dimensional operators. Because the couplings have negative energy dimensions for where operators of dimension-five or more, SMEFT is not Z renormalizable. ϕ ≡ 4 ϕ There are three conditions that SMEFT must satisfy [36]: j d x ðxÞjðxÞ: ð4Þ first, the higher-order operators must satisfy the gauge 3 2 1 symmetries in SM, which is SUð ÞC × SUð ÞL × Uð ÞY. The connected generating functional W½j is defined by Second, the higher-order operators must contain all degrees of freedom of the SM, either basic or composite. Third, at Z½j ≡ expfiW½jg: ð5Þ low energy scales, SMEFT must return to SM provided that there are no weak interacting light particles such as axions The effective action Γ½ϕ¯ is the Legendre transform of or sterile neutrinos. According to Refs. [34,37],wehave W½j as follows: totally 20 dimension-six operators that satisfy these Z requirements and their contributions at one-loop level δW½j Γ½ϕ¯ ¼W½j − d4x jðxÞ; ð6Þ are significant. However, in this paper, we are interested δjðxÞ in the O6 operator of the Higgs potential because it has an important effect to the EWPT process. with

116010-3 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

δW½j ϕ ¼ ϕ¯ ðxÞ ≡ : ð7Þ c δjðxÞ

From Eqs. (6), (7), and (4), it is easy to prove that Z δΓ½ϕ¯ δW½j δj δjðxÞ ¼ − d4x jðxÞþϕ¯ ðxÞ δϕ¯ δj δϕ¯ δϕ¯ δj δj ϕ¯ x − j − ϕ¯ ¼ ð Þ δϕ¯ δϕ¯ ¼ −j: ð8Þ

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 ( P R ∞ in 4 4 Z½j¼ d x1…d x jðx1Þ…jðx ÞG ðx1; …;x Þ n¼0 n! R n n ðnÞ n P n : ð9Þ ∞ i 4 … 4 … c … iW½j¼ n¼0 n! d x1 d xnjðx1Þ jðxnÞGðnÞðx1; ;xnÞ ’ The Green s functions GðnÞ are the sum of all Feynman diagrams with n external lines. The expansion coefficients of iW½j ’ c are connected Green s functions, and GðnÞ 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¯ ðxÞ field as follows: Z X∞ 1 ¯ 4 4 ¯ ¯ ðnÞ Γ½ϕðxÞ ¼ d x1…d x ϕðx1Þ…ϕðx ÞΓ ðx1; …;x Þ: ð10Þ n! n n n n¼0

ΓðnÞ is the sum of all one-particle-irreducible Feynman diagrams with n external lines. The Fourier transform of the effective action in ϕ¯ ðxÞ is h i ( R Q 4 ΓðnÞ … n d pi 2π 4δ 4 ΓðnÞ … ðx1; ;xnÞ¼ i¼1 2π 4 expfipixig ð Þ ð Þðp1 þþpnÞ ðp1; ;pnÞ; R ð Þ ð11Þ ϕ˜ ðpÞ¼ d4xe−ip:xϕ¯ ðxÞ:

From the two Fourier transforms above, the effective action in (10) is rewritten as follows: Z X∞ 1 Yn 4 ¯ d pi ˜ 4 ðnÞ Γ½ϕðxÞ ¼ ϕð−p Þ ð2πÞ δð4Þðp1 þþp ÞΓ ðp1; …;p Þ: ð12Þ n! 2π 4 i n n n¼0 i¼1 ð Þ

The Fourier transform of ϕc is Z ˜ 4 −ip:x 4 ϕcðpÞ¼ d xe ϕc ¼ ϕcð2πÞ δð4ÞðpÞ: ð13Þ

The effective potential becomes Z X∞ 1 Yn 4 d pi ˜ 4 ðnÞ Γ½ϕ ¼ ϕ ð−p Þ ð2πÞ δð4Þðp1 þþp ÞΓ ðp1; …;p Þ c n! 2π 4 c i n n n¼0 i¼1 ð Þ X∞ 1 ϕn 2π 4δ 4 0 ΓðnÞ 0 ¼ ! cð Þ ð Þð Þ ðpi ¼ Þ 0 n Zn¼ X∞ 1 ¼ ϕnΓðnÞðp ¼ 0Þd4x: ð14Þ n! c i n¼0

116010-4 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

TABLE I. The Feynman rules at zero and finite temperature, β is the Matsubara factor [43].

Propagator T ¼ 0 T ≠ 0 i μ 0 i μ −1 Boson 2 2 ; p p ; p 2 2 ; p 2niπβ ; p p −m þiϵ ¼ð Þ p −m þiϵ ¼ð Þ Fermion i ; pμ ¼ðp0; pÞ i ; pμ ¼ðð2n þ 1Þiπβ−1; pÞ p=−mþiϵ p=−mþiϵ R 4 P R 3 Loop integral d p i n¼∞ d p ð2πÞP4 β n¼−∞ ð2πPÞ3 Vertex function 2π 4δ 4 − β 2π 3δ δ 3 p ð Þ ð Þð i piÞ i ð Þ Σωi ð Þð i iÞ

Meanwhile, the definition of the effective potential is 2. The perturbation approach Z First, we consider a toy model describing a self- interacting real scalar field and we see that this scalar field Γ½ϕ ¼− d4xV ðϕ Þ: ð15Þ c eff c can be Higgs field. The scalar field satisfies the following equation of motion [44]: Comparing the above two equations (14) and (15), we get ;α χ;α þ V0ðχÞ¼0; ð17Þ X∞ 1 V ¼ − ϕnΓðnÞ: ð16Þ where V0 ¼ ∂V, potential V is unknown. The field χ can eff n! ∂χ n¼0 always be decomposed into homogeneous and inhomo- geneous components by adding a contribution from thermal The Feynman rules at zero and finite temperature are fluctuation ϕ, shown in Table I [43], which is used to calculate the n Green’s functions Γ or the sum of all diagrams. χðxÞ¼χ¯ðtÞþϕðxÞ: The existence of dimension-six operators will affect many scattering and decay processes happening in particle Using Taylor series for Vðχ¯ þ ϕÞ,wehave accelerators, so the nonzero Wilson coefficients inferred from fitting methods will be the indication of new physics. ∂Vðχ¯Þ ϕ2 ∂2Vðχ¯Þ ϕ3 ∂3Vðχ¯Þ However, most of the Wilson coefficients in the electro- V χ V χ¯ ϕ ð Þ¼ ð Þþ ∂χ þ 2 ∂2χ þ 6 ∂3χ weak sector except c6 are tightly constrained around zero 4 [1]. The difficulty in constraining c6 lies in the fact that the þ Oðϕ Þ: Higgs potential in the SM is conventionally assumed to be 2 4 of the form ∼ϕ þ ϕ , but the Higgs self-coupling λ is ∂ ∂Vðχ¯Þ ∂ϕ 1 ð ∂χ Þ 0 poorly measured. Moreover, with our choice of the form of Noting that ∂χ ¼ and ∂χ ¼ , we get O6, it will not affect λ anyway; in other words, new physics cannot be inferred from the measurement of λ but has ϕ2 0 χ 0 χ¯ ϕ 00 χ¯ 000 χ¯ important effects on EWPT. So it is reasonable to work V ð Þ¼V ð Þþ V ð Þþ 2 V ð Þ: ð18Þ solely with O6 in this paper. We also note that this one-loop effective potential has Substituting (18) into (17), and we only consider the already included some dimension-six operators other than potential component and averaging over space, we obtain O6. For example, the one-loop gauge boson term corre- 2 4 ðnÞ sponding to Or ∼ W ϕ is included in the Γ function ;α 1 2 χ;α V0 χ¯ V000 χ¯ ϕ 0: 19 with n ¼ 4 external lines of the Higgs field. There are some þ ð Þþ2 ð Þh i¼ ð Þ 2 2 exceptions like, for example, OH ∼ ð∂μϕ Þ which cannot be implemented in the calculation of the effective potential The last two components in (19) can be rewritten as the but they have contributions to the sphaleron energy; we will derivative of an effective potential. In scalar field theory, we not consider them in this paper. can get The next work is from the Lagrangian of each field; we Z will calculate all their contributions by summing all the 1 2 1 ϕ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik Green’s functions in Eq. (16) and renormalization. We will h i¼2π2 2 þ nk dk: ð20Þ k2 m2 χ¯ have two components, one is quantum contributions and þ ϕð Þ the other is thermal contributions. The thermal contribution 2 00 is obtained by the finite temperature Matsubara Green’s Taking into account that mϕ ¼ V , we can rewrite the third function (or temperature propagators in Table I). term in (19) as

116010-5 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

2 Z 2 2 Z 2 1 1 ∂mϕðχ¯Þ k dk 1 ∂mϕðχ¯Þ k n dk 000 χ¯ ϕ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik V ð Þh i¼ 2 þ 2 2 8π ∂χ¯ 2 2 4π ∂χ¯ 2 2 mϕðχ¯Þþk mϕðχ¯Þþk 1 2 ∂Vϕ ∂Vϕ ¼ ∂χ¯ þ ∂χ¯ ; where Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k k2 þ m2ð2k2 þ m2Þ − m4 log ½2ðk þ k2 þ m2Þ V1 k2 m2 χ¯ k2dk ; ϕ ¼ 4π2 ϕð Þþ ¼ 32π2 ð21Þ Z ∂V2 1 ∂m2 ðχ¯Þ 2 ϕ ϕ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik nkdk ¼ 2 : ð22Þ ∂χ¯ 4π ∂χ¯ 2 2 mϕðχ¯Þþk

Using Taylor expansion for (21), we obtain

M4 m2M2 m4 m4 m4 1 Vϕ m ¼ 16π2 þ 16π2 þ 128π2 þ 32π2 ln þ 32π2 ln 2 þ M m4 m2 V∞; ¼ 64π2 ln μ2 þ ð23Þ in which M4 m2M2 m4 2M V∞ ¼ þ − ln þ: ð24Þ 16π2 16π2 32π2 e1=4μ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 The occupation numbers nk are given by the Bose-Einstein formula, ωk ¼ k þ mϕ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z ω2 m2 1 ∞ 2 2 ∞ k − ϕ ω 2 k dk T T2 T2 k hϕ i ¼ ω ¼ ω d : T 2 k 2 mϕ k 2π 0 ω T − 1 2π T − 1 T ke T e

1 Jð Þ has the following form: Z Z ∞ 2 − α2 ν=2 ∞ 2 − α2 ν=2 ðνÞ α β ðx Þ ðx Þ J∓ ð ; Þ¼ −β dx þ β dx: ð25Þ α ex ∓ 1 α exþ ∓ 1

1 Substituting ν ¼ 1, we obtain an expansion of J∓, Z ∞ 2 − α2 1=2 ð1Þ α β 0 2 ðx Þ J∓ ð ; ¼ Þ¼ x dx α e ∓ 1 ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 2 2 1 2 α 1 2 3 π − 2 β − π α − β − 2 α ðlnð4πÞþC − 2ÞþOðα Þ; ¼ 1 2 1 2 1 2 α 1 2 ð26Þ 6 π þ 2 β þ 2 α ðlnðπÞþC − 2ÞþOðα Þ:

Therefore, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ω2 m2 2 ∞ k − ϕ ω ϕ2 T T2 T2 k h i ¼ 2 ω d T 2π mϕ k e T − 1 T T 2 ϕ χ¯ T ð1Þ m ð Þ ¼ J− ; 0 ð27Þ 4π2 T

116010-6 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020) and Higgs potential, as mentioned in the functional approach, we have summed up all the contributions of one-loop diagrams. 2 1 ∂ ϕ 000 2 m T 1 So, in the effective potential, there are always the contribu- V ðχ¯Þhϕ i ¼ mϕ J− 2 T ∂χ¯ 4π2 tions of some dimension-six operators. We did not consider T daisy loops, but we know that the thermal contributions of ∂Vϕ daisy loops are quite small, they can be ignored; this was also ¼ ∂χ¯ ; ð28Þ mentioned in the Introduction section. Finally, when calcu- with lating the effective potential to investigate the EWPT and the sphaleron, we only need to use the O6 operator, regardless of Z 4 mϕ 4 the other dimension-six operators. T T T mϕ T α 1 α 0 α Vϕ ¼ 2 J−ð ; Þd ¼ 2 F− : ð29Þ 4π 0 4π T B. Effective potential at T =0 α mϕ In case ¼ T ,wehave The O6 operator is an addition to the Higgs potential, with all the contributions of SM bosons and top quark, ∂ T ∂ T ∂ T Vϕ Vϕ ∂α Vϕ ∂mϕ Eq. (16) becomes ¼ ¼ ; ð30Þ ∂χ¯ ∂α ∂χ¯ ∂α T∂χ¯ ϕ 0 ϕ 0 ϕ 0 O Veffð ; Þ¼V0ð ; ÞþV1ð ; Þþ 6 Substituting Eq. (30) into Eq. (29) yields m2 λ Z ¼ Λ − R ϕ2 þ R ϕ4 T 3 mϕ 1 R 2 4 ∂Vϕ ∂mϕ ∂ α α 0 T T ½ J−ð ; Þ α ¼ 2 d 1 X m2 ϕ ∂χ¯ ∂χ¯ 4π 0 ∂α 4 ϕ i ð Þ þ 2 nimi ð Þ ln 2 2 64π v ∂mϕ i¼h;W;Z;t T 1 α 0 ¼ ∂χ¯ 4π2 mϕJ−ð ; Þ: ð31Þ 1 ϕ2 − v2 3: þ 8Λ2 ð Þ So Eq. (31) is equivalent to Eq. (28). V∞ can be absorbed by a redefinition of constant in the The O6 operator was carefully chosen so that it does not original potential. The potential V is affect the parameters and mass of the Higgs boson.

2 Specifically, λ0 4 m0 2 Vðχ¯Þ¼ χ¯ þ χ¯ þ Λ0; ð32Þ 8 4 2 > > O6 ¼ 0; > ϕ <> ¼v The final result, which includes both quantum and dO6 6c6 ϕ ϕ2 − 2 2 0 thermal contributions, is ϕ ¼ 8Λ2 ð v Þ ¼ ; > d ϕ¼v ϕ¼v > 4 2 4 > 2O 6 mϕðχ¯Þ mϕðχ¯Þ mϕ > d 6 c6 2 2 2 2 2 2 T : 2 ¼ 2 ½ðϕ − v Þ þ 4ϕ ðϕ − v Þ ¼ 0: V ¼ V þ ln þ F− : ð33Þ dϕ ϕ 8Λ ϕ eff 64π2 μ2 4π2 T ¼v ¼v ð35Þ At zero temperature, the last term in (33) will vanish. Then the effective potential reduces to The normalization conditions for such the potential are 4 χ¯ 2 χ¯ 8 mϕð Þ mϕð Þ 0 0 VT¼0 V ln : 34 < Veffðv; Þ¼ ; eff ¼ þ 64π2 μ2 ð Þ V0 v; 0 0; : effð Þ¼ ð36Þ 00 0 2 Therefore, if we calculate the contributions of other Veffðv; Þ¼mh: particles and perform the renormalization process, we get the full one-loop effective potential. From these conditions, expanding the logarithmic func- In addition, we have another way when the gauge and tions, we get ring loops are taken into account by using functional h   i 8 2 P 2 ℏ mh 1 4 mi 3 methods, Nielsen identities, and the small expand as > λ ¼ 2 − 2 4 n m ln 2 þ ; > R 2v 16π v i i v 2 in papers [18,19]. However, we also see that the gauge and > i¼h;W;Z;t < 2 P ring loops do not play a major role in the phase transition 2 mh 1 4 m ¼ − 2 2 n m ; problem. > R 2 16π v i i ð37Þ > i¼h;W;Z;t Reference [37] has shown the contributions of the > 2 2 P > m v 1 : Λ h − 4 dimension-six operators in the energy of sphaleron at R ¼ 8 128π2 nimi : one-loop level. In the calculation of the one-loop effective i¼h;W;Z;t

116010-7 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

C. Effective potential at T ≠ 0

The effective potential of SM when adding the O6 operator and expanding Eq. (16) [or Eq. (33)] in detail is ϕ ϕ 0 ϕ O Veffð ;TÞ¼Veffð ; ÞþV1ð ;TÞþ 6 λ ðTÞ ϕ4 − ϕ3 2 − 2 ϕ2 ¼ 4 ET þ DðT T0Þ 1 Λ T ϕ2 − v2 3; þ ð Þþ8Λ2 ð Þ ð38Þ where 1 X a T2 λðTÞ ≡ λ þ n m4 ln i ; R 16π2v4 i i v2 i¼h;W;Z X 3 3 3 1 m þ 6m þ 3m E ≡ n m3 h W Z ; 12π 3 i i ¼ 12π 3 v i¼h;W;Z v X 2 2 2 2 2 2 FIG. 1. The vacuum contribution to effective potential. Dark n m n m m þ 6m þ 3m þ 6m D ≡ i i − t t ¼ h W Z t ; line in blue: SM. Dashed line in orange: SM+ O6. 24v2 48v2 24v2 i¼h;W;Z 2 2 mR The vacuum contribution of the effective potential is T0 ≡ ; Λ 800 2D depicted in Fig. 1 with ¼ GeV. The vacuum con- X 2 4 2 4 2 4 0 n π T 7π T 41π T tribution (T ¼ ) of SMEFT increases more rapidly and is ΛðTÞ ≡ Λ − i þ n ¼ Λ − : ϕ 246 R 90 t 720 R 180 larger than that of the SM in the range > GeV. In i¼h;W;Z contrast, the behavior of the potentials in the range ϕ < 246 GeV is opposite to that in the range ϕ > 246 GeV. Using Eq. (37), we obtain the explicit expressions of the This suggests that the minimum of SMEFT is more stable normalization parameters which was derived from the one- than that of the SM. loop effective potential at zero temperature, as follows:

m2 1 X m2 3 1 X a T2 λðTÞ¼ h − n m4 ln i þ þ n m4 ln i 2v2 16π2v4 i i v2 2 16π2v4 i i v2 i¼h;W;Z;t i¼h;W;Z 2 1 X 2 mh − 4 mi ¼ 2 2 4 nimi ln 2 ; 2v 16π v AiT i¼h;W;Z;t 1 m2 1 X 1 m2 1 X T2 ¼ h − n m4 ¼ h − n m4 ; 0 2D 2 16π2v2 i i D 4 32π2v2 i i i¼h;W;Z;t i¼h;W;Z;t 2 2 1 X 41π2 4 Λ mhv − 4 − T ðTÞ¼ 8 128π2 nimi 180 : i¼h;W;Z;t

Here, we have put The above expansion for a high-temperature effective  potential will be better than 5% if mboson < 2.2 [45], where ≡ − 3 2 3 9076 T Boson : lnðABÞ lnðabÞ = ¼ . ; m is the relevant boson mass. The mass range of SM ð39Þ boson Fermion : lnðAFÞ ≡ lnðafÞ − 3=2 ¼ 1.1351: particle in the EWPT calculations always fit with this because the critical temperature is about 100 GeV. The transition-ending temperature is the temperature at The effective potential is drawn in Fig. 2. It appears the which the initial minimum becomes a maximum, nonzero second minimum and the value of this minimum grows as temperature decreases. Between two minima, a 3 4 02 2 v barrier appears which is a sign of first-order phase T0 ¼ T0 − : 8DΛ2 transition.

116010-8 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

2 O μ λ c6 SMþ 6 ϕ Ω − ϕ2 ϕ4 ϕ2 − 2 3 Vtree ð Þ¼ 2 þ 4 þ 8Λ2 ð v Þ ;

where Ω is a cosmological constant and λ > 0. Here we have temporarily written down explicitly the Wilson coefficient of O6 since we will also have the chance to discuss its sign. We find the local extremes of potential by calculating its first derivative

3c6 V0 ϕ −μ2ϕ λϕ3 ϕ ϕ2 − v2 2 0: ð Þ¼ þ þ 4Λ2 ð Þ ¼  ϕ ¼ 0; ⇒ ; ð40Þ Aϕ4 þðλ − 2Av2Þϕ2 − μ2 þ Av4 ¼ 0

A ≡ 3c6 here we have set 4Λ2 for brevity, and the discriminant Δ of the quartic equation is

Δ ¼ λ2 − 4Aλv2 þ 4Aμ2: ð41Þ

On the other hand, since O6 does not affect three normal- 0 0 0 00 2 FIG. 2. The effective potential at finite temperature in SM with ized conditions: VðvÞ¼ ;V ðvÞ¼ ;V ðvÞ¼mh so the O6 (Λ ¼ 800 GeV). Blue line: T ¼ 120 GeV. Orange line: coefficients in the Higgs potential are still the same form as 119 35 118 56 T ¼ T1 ¼ . GeV. Green line: T ¼ Tc ¼ . GeV. those of SM, 0 Red line: T ¼ 118 GeV. Purple line: T ¼ T0 ¼ 113.36 GeV 8 2 2 (the end of phase transition). < μ ¼ λv ; m2 2λv2; : h ¼ ð42Þ The results of phase transition in the range 700 GeV ≤ 1 4 Ω ¼ 4 λv : Λ ≤ 1000 GeV are summarized in Table II. An interesting 2 2 conclusion is that when Λ increases, T1 (the second Substituting μ ¼ λv into Eq. (41), we get minimum starts to appear from here) and the critical Δ λ2 0 temperature (Tc) approach each other. As expected, the ¼ > : bigger the cutoff scale, the smaller the phase transition strength. When Λ → ∞, the transition strength returns to The solutions of the quartic equation are the value of the SM. The maximum possible value of the  2A 2 − λ λ ϕ2 ¼ v2; cutoff scale at which the phase transition strength is greater 2 v ϕ ¼ → 2 : 2A 2 Av −λ than unity is about 860 GeV. ϕ ¼ A At zero temperature, we expect that Eq. (40) should have Λ D. The lower bound of three solutions corresponding to the three extremes at ϕ ¼ From pure mathematical arguments, we can also find the 0 and ϕ ¼v, so we must have the following condition: lower bound of Λ. The tree-level Higgs potential in SM Av2 − λ with the inclusion of O6 is 0 A < :

m2 TABLE II. Phase transition strengths with different values of Λ. λ h From Eq. (42), with ¼ 2v2, we get Λ (GeV) T1 (GeV) Tc (GeV) vc (GeV) vc=Tc λ A < ; 1000 132.89 132.82 86.53 0.65 v2 900 127.07 126.68 114.22 0.9 3 λ 2 ⇒ c6 mh 880 125.74 125.24 120.79 0.96 2 < 2 ¼ 4 ; 870 125.08 124.48 124.38 0.999 4Λ v 2v 4 860 124.08 123.71 127.81 1.03 3c6v ⇒ Λ2 > ; 840 122.49 122.08 135.21 1.11 2m2 820 120.97 120.38 142.36 1.18 h pffiffiffiffiffiffiffiffiffiffiffi 2 800 119.35 118.56 149.78 1.26 ⇒ Λ 1 5 v 700 111.02 107.53 184.72 1.72 > . c6 : ð43Þ mh

116010-9 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

GeV difficulty is that it is unclear whether the Wilson parameters 1200 are positive or negative; the second difficulty is that the

1000 equations of motion are temperature dependent when considering EWPT and hence are even more difficult to 800 be solved.

600 A. Sphaleron Ansatz

400 Klinkhamer and Manton [46] have found the Ansatz approach to find an approximate sphaleron solution without 200 fully solving all the equations of motion. The contribution to the sphaleron energy of the hypercharge gauge field is C6 H 0.5 1.0 1.5 2.0 found to be very small, so we will not consider them (i.e., Bμ ¼ 0) [46]. We also choose the temporal gauge, so FIG. 3. The shaded area is the allowed range for the cutoff a xi W0 ¼ 0. Let ni ≡ where r is the radial coordinate in the parameter according to the Wilson parameter c6. r spherical coordinates and assume that sphaleron has a spherically symmetric form. The sphaleron Ansätze then We see that the case c6 < 0 must be removed. With have the following forms [34]: c6 ¼ 1 and substituting the experimental values v ¼ 8 246 GeV and m ¼ 125 GeV into (43), we get Λ > > 0 h < pvffiffi σa 593 HðrÞ¼ 2 hðrÞina ; GeV. Therefore, the larger the Wilson parameter, 1 ð45Þ the bigger the lower bound. Overall, we have > : a 2 ϵaij fðrÞ Wi ðrÞ¼g nj r ; pffiffiffiffiffiffiffiffiffiffiffi v2 pffiffiffiffiffi Λ 1.5c6 < Λ < 860: c6 ⇔ 593 GeV < pffiffiffiffiffi with the boundary conditions m c6 h  < 860 GeV: ð44Þ hðr → 0Þ¼fðr → 0Þ¼0; ð46Þ hðr → ∞Þ¼fðr → ∞Þ¼1: Thus, the larger the Wilson parameter value, the wider the domain of Λ in order to have the first-order phase The functions fðrÞ and hðrÞ are called radial functions (or transition, as shown in Fig. 3. profile functions). The first boundary condition is intended to avoid singularity at r ¼ 0, and the second boundary condition is to ensure that the Ansätze are asymptotic to the IV. THE ENERGY OF SPHALERON form of the fields at infinity [46]. As we will see, in this way In a non-Abelian gauge theory, such as the electroweak the equations of motion will be converted to the equations theory, there is another interesting kind of transition called of hðrÞ and fðrÞ and everything will be much simpler. topological phase transition, which is the transition of the Next, in order to calculate the sphaleron energy, we need field configurations between topologically distinct vacua. to solve all the equations of motion to find hðrÞ and fðrÞ When such a transition occurs, the baryon number will be and then substitute these functions into the sphaleron violated and it opens up the possibility for the electroweak energy functional. However, solving these equations of theory to satisfy the first condition of Sakharov. At zero motion is also very complicated and can only be done by temperature, the instanton process is strongly suppressed numerical method as in Ref. [34]. To be more streamlined, due to the smallness of the weak coupling [44]. On the we can assume the above profiles by spherical functions other hand, at high temperature, the sphaleron process and refine these functions through free parameters. The dominates and the transition rate is proportional to the calculation steps are shown in the next section. Γ ∼ … − Boltzmann factor as sph OðT; Þ expð Esph=TÞ. When the temperature is high, the “height” of the potential, which B. Contributions to the sphaleron energy is the energy of sphaleron, is small and thermal equilibrium The sphaleron energy functional takes the form can be obtained. However, when the temperature is low, the Z 1 height of the potential barrier will rise and the sphaleron 3 † E ¼ d x Wa Wa þðD HÞ ðD HÞþV ðH;TÞ : rate must be strongly suppressed. The universe also sph 4 kl kl k k Higgs expands while this phase transition occurs, so the condition Γ ≪ ð47Þ for the departure from thermal equilibrium is sph Hrad. According to Ref. [34], the sphaleron energy at zero (i) The first term in Eq. (47) is the contribution of the 1 a a;kl temperature at tree level is calculated by including all isospin gauge fields 4 WklW : Substituting the relevant dimension-six operators. However, the first Ansätze in (45) into the first term in (47) yields

116010-10 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

1 2 a a;kl 2 02 4 4 2 2 1 − 2 C. Profile functions WklW ¼ 2 6 ½ f r þ r f ð fÞ : 4 g r There are two kinds of Ansatz: the Ansatz with scale-free parameters first introduced in [46] and the smooth Ansatz So, the sphaleron energy contribution of the gauge motivated from kink-type solutions [34,47]. We will use fields is both these two types to calculate the sphaleron energy at Z zero temperature and compare it to the values obtained 4π ∞ 8f2ð1 − fÞ2 gauge 4 02 from numerical calculations, then we will use the better Esph ¼ 2 dr f þ 2 : ð48Þ g 0 r Ansatz to calculate sphaleron energy at finite temperature. We use the following Ansatz: We can use the following dimensionless distance (1) Ansatz a: f and h with scale-free parameters, variable: ( ξ2 ξ ≤ ξ 2 2 ; a; df df d df a ξ a ξ ≡ gvr → dξ gvdr → gv : f ð Þ¼ 2 ð52Þ ¼ ¼ ξ ¼ ξ 1 − a ξ ≥ dr d dr d 2ξ2 ; a; ( 4ξ So, we get the sphaleron energy of the gauge fields 5 ; ξ ≤ b; a b in terms of the variable ξ as follows: h ðξÞ¼ 4 ð53Þ 1 − b ξ ≥ Z 5ξ4 b; 4π ∞ 2 8 2 1 − 2 gauge v ξ 4 df f ð fÞ Esph ¼ d þ 2 : with a, b are scale-free parameters. g 0 dξ ξ (2) Ansatz b: f and h are smooth functions, ð49Þ fbðξÞ¼tan hðAξÞ; hbðξÞ¼tan hðBξÞ; ð54Þ (ii) The second term in Eq. (47) is the kinematic of the † where A and B are free parameters. Higgs field ðDkHÞ ðDkHÞ. Following the same procedure as we did in the Note that we must select the profile functions that satisfy previous case, the contribution of the kinematic term the boundary conditions in (46). of the Higgs field to the sphaleron energy is Z D. Sphaleron energy at zero temperature 4π ∞ O kinematic Higgs v ξ at tree level with 6 Esph ¼ d g 0 The zero temperature sphaleron energy at tree level with 2 ξ dh 2 O6 is × þ h2ð1 − fÞ2 : ð50Þ Z 2 dξ 4πv ∞ df 2 8 tree ξ 4 2 1 − 2 Esph;0 ¼ d þ 2 f ð fÞ g 0 dξ ξ (iii) The third term in Eq. (47) is the Higgs potential with 2 2 O6 at tree level, ξ dh þ þ h2ð1 − fÞ2 2 dξ 0 VHiggsðH; T ¼ Þ 2 2 ξ λ 2 2 v 2 3 1 2 3 þ ðh − 1Þ þ ðh − 1Þ : ð55Þ −μ2 † λ † 2 Ω † − v g2 4 8Λ2 ¼ ðH HÞþ ðH HÞ þ þ Λ2 H H 2 As mentioned in the Introduction, the Wilson coefficient of μ2v2 λv4 v6 − 2 4 Ω 2 − 1 3 O6 can be absorbed into Λ. Assuming that the Higgs ¼ h þ h þ þ 2 ðh Þ 2 4 8Λ potential in the SM has the standard form ∼ϕ2 þ ϕ4,we 4 6 λv v have λ ≈ 0.1. Because choosing the O6 operator does not h2 − 1 2 h2 − 1 3: ¼ 4 ð Þ þ 8Λ2 ð Þ ð51Þ change the Higgs mass form, so we still have the value of λ like in the SM. It is emphasized that, according to Ref. [21], the (1) Ansatz a [Eqs. (52) and (53)] contributions to the sphaleron energy from the first and With Ansatz a, Eq. (55) becomes the second terms in Eqs. (49) and (50), respectively, are 832 104a4 704a3 large. The difference between sphaleron energy at zero Etree − sph;0 ¼ 315 1155 3 þ 2625 2 temperature and sphaleron energy at finite temperature lies a b b solely in the Higgs potential. The form of contributions 4 b 1657500 from kinematic terms remains the same, but the Higgs þ 43509375 profile function hðrÞ and the dimensionless distance must 2 be rescaled when calculating sphaleron energy at nonzero b 447100λΛ2 − 137997 2 þ 2 2 ð v Þ : ð56Þ temperature. g Λ

116010-11 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

TABLE III. Sphaleron energy at zero temperature with O6 in Ansatz a, in units of 4πv=g ≈ 4, 738 TeV.

Λ tree 4πv [GeV] Esph;0½× g TeV ab ∞ 2.024 2.36 2.23 1000 2.002 2.39 2.33 900 1.997 2.41 2.36 800 1.989 2.42 2.40 700 1.976 2.45 2.47

TABLE IV. Sphaleron energy at zero temperature with O6 in Ansatz b, in units of 4πv=g ≈ 4, 738 TeV. FIG. 4. The tree-level profile functions for the Higgs field and Λ tree 4πv the isospin gauge fields at zero temperature using Ansatz b with [GeV] Esph;0½× g TeV AB Λ ¼ 800 GeV. Orange line: the Higgs field. Blue line: the isospin ∞ 2.144 0.22 0.48 gauge fields. 1000 2.126 0.22 0.47 900 2.122 0.21 0.45 800 2.115 0.21 0.44 only about 5.6%, while Ansatz b gives a difference of about 700 2.104 0.21 0.43 11.9%. In addition, the sphaleron energy which has the contribution of O6 at zero temperature, with Λ ranging from 590 to 860 GeV, is in the range ½1.8–2 × 4πv ðTeVÞ To minimize this energy according to the parameters g a and b, we get the energy value table of sphaleron [34]. So, according to Tables III and IV, Ansatz a also gives energy in the cutoff scale as Table III. better results. Therefore, we will only use Ansatz a to (2) Ansatz b [Eq. (54)] calculate the sphaleron energy at nonzero temperature at With Ansatz b, we get the energy value table of one-loop level. sphaleron energy in the cutoff scale as Table IV. The profile functions of Ansatz b are depicted in Fig. 4. E. Sphaleron energy at finite temperature With Ansatz a, the sphaleron energy in the SM (Λ ¼ ∞) at one-loop level with O6 2 024 4πv is about . × g ðTeVÞ. Comparing this value to the Substituting the one-loop effective potential [Eq. (38)] value obtained from numerical method in Ref. [34], we find into the sphaleron energy functional, we obtain the spha- that Ansatz a is in good agreement and the difference is leron energy at the temperature T,

Z 4π ∞ 2 8 ξ2 2 ξ2 1−loop v ξ 4 df 2 1 − 2 dh 2 1 − 2 Esph ðTÞ¼ d þ 2 f ð fÞ þ þ h ð fÞ þ 2 4 Veffðh; TÞ : g 0 dξ ξ 2 dξ g v

Taking the variations with respect to f and h, we get the we want to express it through the profile functions h and f. following system of equations of motion: From the Ansatz of the Higgs doublet and gauge field in 8 Eq. (45), we obtain < ξ2 d2f 2 1 − 1 − 2 − ξ2 2 1 − dξ2 ¼ fð fÞð fÞ 4 h ð fÞ : 57 v2h2 ξ2 ∂ ð Þ † : d ξ2 dh 2 1 − 2 Veff ðh;TÞ H H ¼ : dξ ð dξÞ¼ hð fÞ þ g2v4 ∂h 2

The profile functions do not necessarily satisfy the On the other hand, ignoring the contribution of the nonlinear equations Eq. (57). However, as we shall see, Goldstone bosons, the Higgs doublet is finding an Ansatz satisfying the equations of motion at a certain asymptotic limit, especially in the sphaleron core 1 0 ϕ2 H ¼ pffiffiffi → H†H ¼ : region, will give better results. The “Ansatz a” satisfies 2 ϕ 2 these equations in the sphaleron core region (ξ → 0), and Ansatz this is also suitable to calculate sphaleron energy at So, we have the following relationship: finite temperature. In the electroweak phase transition in Sec. III, the ϕ effective potential depends on ϕ. In the sphaleron problem, ¼ vh:

116010-12 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

The above equation allows us to convert the effective ξ vðTÞ ξ˜ ¼ ; h˜ ¼ sh; with s ≡ : potential in ϕ to that in h. s v 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 To avoid cumbersome, we will ignore the tilde symbol should be noted that we must rescale the variable ξ and the and automatically perform the above operation. The effec- radial function hðξÞ as follows: tive potential is then given by

λ 1 ðTÞ ϕ4 − ϕ3 2 − 2 ϕ2 Λ ϕ2 − 2 3 Veffðh; TÞ¼ 4 ET þ DðT T0Þ þ ðTÞþ8Λ2 ð v Þ λðTÞv4 v6 s4h4 − ETv3s3h3 D T2 − T2 v2s2h2 s2h2 − 1 3 Λ T ¼ 4 þ ð 0Þ þ 8Λ2 ð Þ þ ð Þ λ 4 ðTÞv 4 4 − 1 − 3 3 3 − 1 2 − 2 2 2 2 − 1 ¼ 4 s ðh Þ ETv s ðh ÞþDðT T0Þv s ðh Þ 6 v 6 6 − 1 − 3 4 4 − 1 3 2 2 − 1 þ 8Λ2 ½ðs ðh Þ s ðh Þþ s ðh Þ þ CðTÞ: In the above equation, in the first line, we only rewrite the effective potential in Sec. III; in the second line, we substitute ϕ ¼ vh and rescale the radial function h; in the third line, we add and remove the temperature-dependent constants and set it to CðTÞ. The sphaleron energy will be rewritten as Z  4π ∞ 2 8 ξ2 2 v ξ 2 4 df 2 1 − 2 dh 2 1 − 2 EsphðTÞ¼ d s þ 2 f ð fÞ þ þ h ð fÞ gs 0 dξ ξ 2 dξ ξ2 λðTÞ ET DðT2 − T2Þ þ s4ðh4 − 1Þ − s3ðh3 − 1Þþ 0 s2ðh2 − 1Þ g2s2 4 v v2  v2 s6 h6 − 1 − 3s4 h4 − 1 3s2 h2 − 1 : þ 8Λ2 ½ð ð Þ ð Þþ ð Þ

The constant CðTÞ is removed so that the sphaleron showed in the last column the sphaleron energy calculated energy does not diverge. This is completely normal because from the well-known scaling expression [21,22,48], this cosmological constant can be absorbed into the three normalized conditions of the effective potential. In order to vðTÞ E ðTÞ ≈ E ðT ¼ 0Þ: ð58Þ ensure the convergence of sphaleron energy, we deliber- scaling v sph ately chose the Ansatz hðξÞ taking the form ∼ξ−4 in the In [34], the authors only used the above approximate limit ξ → ∞, so that after integrating we get the form ∼ξ−1. expression and did not calculate directly the sphaleron The result is given in Table V. We see that the sphaleron energy at nonzero temperature from the effective potential. energy decreases with increasing temperature. This is The profile functions of sphaleron for the Higgs boson consistent with our expectation of the large baryon viola- and isospin gauge fields are plotted in Figs. 5 and 6, tion rate in the early universe. For comparison purposes, we respectively.

TABLE V. Sphaleron energy at finite temperature using Ansatz a, in units of 4πv=g ≈ 4, 738 TeV with Λ ¼ 800 GeV.

T [GeV] vðTÞ [GeV] EðTÞ abEscalingðTÞ (% deviation)

T1 ¼ 119.35 138.77 1.035 2.853 3.612 1.122 (8.41%) 119 144.47 1.087 2.785 3.411 1.168 (7.45%) Tc ¼ 118.56 149.78 1.136 2.738 3.272 1.211 (6.60%) 117 162.32 1.249 2.646 3.008 1.312 (5.04%) 116 168.02 1.301 2.617 2.927 1.359 (4.46%) 115 172.78 1.344 2.591 2.855 1.397 (3.94%) 0 T0 ¼ 113.36 179.26 1.404 2.563 2.779 1.449 (3.21%)

116010-13 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

TABLE VI. fTc;vc;EðTcÞg at different cutoff values. Λ T v 4πv ab [GeV] c [GeV] c [GeV] EðTcÞ× g [TeV] 900 126.68 114.22 0.862 2.765 3.350 880 125.24 120.79 0.912 2.764 3.349 870 124.48 124.38 0.940 2.763 3.346 860 123.71 127.81 0.966 2.762 3.342 840 122.08 135.21 1.023 2.754 3.319 820 120.38 142.36 1.078 2.748 3.301 800 118.56 149.78 1.136 2.738 3.272

The authors in Ref. [34] carefully calculated the baryon number violation rate and gave results from the condition Γ ≪ FIG. 5. The radial functions of sphaleron for the Higgs field sph Hrad as using Ansatz a with Λ ¼ 800 GeV. Red line: T ¼ Tc. Blue line: T ¼ T0 . Black line: T ¼ 0. EsphðTÞ vðTÞ T 0 − 7 ln þ ln > ð35.9–42.8Þ: T T 100 GeV ð59Þ

We see that the triples fT;vðTÞ;EðTÞg at any temper- atures in Table V satisfy this condition. So, at the cutoff scale of 800 GeV, the baryon number violation is preserved during the phase transition process. In other words, the condition of thermal imbalance has been satisfied. Now we want to further evaluate the feasibility of the decoupling condition according to different cutoff values. We still use the Ansatz a and the sphaleron energy has been calculated at the critical temperature Tc as in Table VI.

FIG. 6. The radial function of sphaleron for the isospin gauge field using Ansatz a with Λ ¼ 800 GeV. Red line: T ¼ Tc. Black line: T ¼ 0.

The sphaleron solutions are only physically meaningful at T ≤ Tc, although these solutions can be constructed mathematically up to T1. Unlike the bubble solution, sphaleron solutions 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 FIG. 7. Description of the decoupling condition. Red zone: the region that does not satisfy the decoupling condition. Yellow T T0 the temperature drops from c to ). On the other hand, zone: the error area corresponding to the right-hand side of EWPT occurs after inflation and belongs to the radiation- Eq. (59). Green zone: the region that satisfies the decoupling 4π3 2 g 4 dominated period so that the Hubble rate is Hrad ¼ 45 2 T . condition. mpl

116010-14 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

From the sets of values in Table VI, the decoupling 5% of the sphaleron energy of SM. If other dimension-six condition (59) is visually depicted in Fig. 7. operators are also constrained in this domain and all of In summary, by calculating the sphaleron energy at finite them have positive contributions, the total contribution will temperature directly from the effective potential, we can be about 35%; this is a very significant contribution that give a more thorough assessment of the thermal imbalance will make the sphaleron energy increase greatly. condition, which is commonly known through vc=Tc > 1. In the event that contributions of all dimension-six This condition is only inherently reliable in the SM. In the operators can compensate for each other, the sphaleron previous section, we saw that the upper bound of the cutoff energy can only slightly increase compared to the SM. In a scale is 860 GeV to satisfy vc=Tc > 1. By assessing the nutshell, there is a need for more information about the decoupling condition when calculating sphaleron energy, dimension-six operators by considering them in other we see that this upper bound is correct. quantum corrections or in the decay channels, which will allow us to calculate the sphaleron energy in more detail. In V. CONCLUSION AND DISCUSSION contrast, our range of the cutoff parameter and the EWPT With the same calculation as in Sec. IV, we can calculate and sphaleron problem are the good binding channels for finding new physics when adding dimension-six operators. the sphaleron energy with the contributions from other 8 dimension-six operators as in Ref. [34]. We also estimate Also, the dimension-eight operator H has been inves- that our calculation has an error not exceeding 10% tigated in Ref. [49], while the UV completions have been compared to the result calculated in Ref. [34]. studied in Refs. [25,49]. These are also our upcoming These results are detailed and also consistent with the calculations as well as checking these solutions with the conclusions in the 2004 paper [31] that explored the EWPT CosmoTransition package [50]. problem in the SM with the O6 operator at one loop. In this Although numerical methods can be used to calculate the paper, the exact value of the Higgs mass is used to find the sphaleron energy without approximations for J∓,we upper bound of Λ (about 860 GeV) in order to satisfy the cannot know the form of the effective potential. We do out-of-thermal equilibrium condition. It also shows the not know if it is minimal or not, and also vðTÞ cannot be narrow temperature range where phase transition occurs. determined, so integrals can easily be diverged and give Among many possible dimension-six operators, we only nonphysical results. It is difficult to control these diver- consider the O6 operator for the Higgs potential. This is not gences by a numerical method. Therefore, if we use a sufficient condition but a prerequisite for explaining numerical methods without using approximations for J, matter-antimatter asymmetry in the universe. Specifically, we need a complex process to ensure the stability and when the EWPT survey had O6, we found that the phase physics of the solution. transition strength (vc=Tc) is large enough to cause thermal In the calculation process of this paper, the Ansatz awas nonequilibrium. In addition, when investigating the spha- carefully chosen to ensure the convergence of the sphaleron leron solution, the baryon number violations are preserved energy at finite temperature at one-loop level, providing during phase transition. Thus, the first and third conditions that the effective potential takes the polynomial form. This of Sakharov are feasible when considering SM þ O6.To form can only be obtained if we use the high-temperature fully solve the problem of matter-antimatter asymmetry, we approximation. If we did not use this approximation, the need to cleverly combine the O6 operator with other effective potential would have a very complicated form and dimension-six operators that cause C and CP violations. therefore the convergence of the sphaleron energy would That is our possible future work. not be guaranteed anymore. We do not know if the contributions of these dimension- These results are a continuation of the previous papers, six operators are positive or negative, so we cannot show a comprehensive view and more complete results for conclude whether the sphaleron energy when calculating the cutoff parameter as well as the finite temperature the addition of all dimension-six operators strongly satisfies electroweak sphaleron in the appropriate Ansäzte. It helps the decoupling condition. However, in order to have a first- to show that the dimension-six operators can be one of the order phase transition, our calculation results show that the good improvements. contribution from the dimension-six Higgs operator must be positive. This opens up the need to study Wilson’s ACKNOWLEDGMENTS parameters more. With the numerical result in [34], we find that in the This research is funded by Vietnam National Foundation allowed range of the cutoff scale (44) the contribution of O6 for Science and Technology Development under Grant to the sphaleron energy at zero temperature does not exceed No. 103.01-2019.346.

116010-15 PHONG, KHIEM, LOC, and LONG PHYS. REV. D 101, 116010 (2020)

[1] F. Goertz, A. Papaefstathiou, L. L. Yang, and J. Zurita, J. [20] S. A. R. Ellis, S. Ipek, and G. White, J. High Energy Phys. High Energy Phys. 04 (2015) 167; C. Englert, R. Kogler, H. 08 (2019) 002; J. De Vries, M. Postma, and J. van de Vis, Schulz, and M. Spannowsky, Eur. Phys. J. C 76, 393 (2016). J. High Energy Phys. 04 (2019) 024; J. de Vries, M. Postma, [2] M. I. Dine, R. G. Leigh, P. Huet, A. Linde, and D. Linde, J. van de Vis, and G. White, J. High Energy Phys. 01 (2018) Phys. Rev. D 46, 550 (1992). 089; C. Balazs, G. White, and J. Yue, J. High Energy Phys. [3] K. Kajantie, M. Laine, K. Rummukainen, and M. 03 (2017) 030. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996); F. Csikor, [21] A. Ahriche, Phys. Rev. D 75, 083522 (2007). Z. Fodor, and J. Heitger, Phys. Rev. Lett. 82, 21 (1999);M. [22] A. Ahriche, Eur. Phys. J. C 66, 333 (2010); A. Ahriche, D’Onofrio, K. Rummukainen, and A. Tranberg, J. High T. A. Chowdhury, and S. Nasri, J. High Energy Phys. 11 Energy Phys. 08 (2012) 123. (2014) 096. [4] M. D’Onofrio, K. Rummukainen, and A. Tranberg, Phys. [23] A. Ahriche and S. Nasri, J. Cosmol. Astropart. Phys. 07 Rev. Lett. 113, 141602 (2014). (2013) 035; A. Ahriche, G. Faisel, S. Y. Ho, S. Nasri, and J. [5] V. Q. Phong, V. T. Van, and H. N. Long, Phys. Rev. D 88, Tandean, Phys. Rev. D 92, 035020 (2015); A. Ahriche, K. L. 096009 (2003). McDonald, and S. Nasri, Phys. Rev. D 92, 095020 (2015); [6] M. Bastero-Gil, C. Hugonie, S. F. King, D. P. Roy, and S. A. Ahriche, S. M. Boucenna, and S. Nasri, Phys. Rev. D 93, Vempati, Phys. Lett. B 489, 359 (2000); A. Menon, D. E. 075036 (2016); A. Ahriche, K. Hashino, S. Kanemura, and Morrissey, and C. E. M. Wagner, Phys. Rev. D 70, 035005 S. Nasri, Phys. Lett. B 789, 119 (2019). (2004); S. W. Ham, S. K. Oh, C. M. Kim, E. J. Yoo, and D. [24] K. Fuyuto and E. Senaha, Phys. Rev. D 90, 015015 (2014); Son, Phys. Rev. D 70, 075001 (2004). Phys. Lett. B 747, 152 (2015); K. Funakubo and E. Senaha, [7] J. M. Cline, G. Laporte, H. Yamashita, and S. Kraml, J. High Phys. Rev. D 79, 115024 (2009). Energy Phys. 07 (2009) 040. [25] M. Spannowsky and C. Tamarit, Phys. Rev. D 95, 015006 [8] S. Kanemura, Y. Okada, and E. Senaha, Phys. Lett. B 606, (2017). 361 (2005); G. C. Dorsch, S. J. Huber, and J. M. No, J. High [26] V. Q. Phong, H. N. Long, V. T. Van, and L. H. Minh, Eur. Energy Phys. 10 (2013) 029. Phys. J. C 75, 342 (2015). [9] S. W. Ham, S.-A. Shim, and S. K. Oh, Phys. Rev. D 81, [27] C. W. Chiang and T. Yamada, Phys. Lett. B 735, 295 (2014). 055015 (2010). [28] V. Q. Phong, N. C. Thao, and H. N. Long, Phys. Rev. D 97, [10] S. Das, P. J. Fox, A. Kumar, and N. Weiner, J. High Energy 115008 (2018). Phys. 11 (2010) 108; D. Chung and A. J. Long, Phys. Rev. [29] J. R. Espinosa, T. Konstandin, J. M. No, and M. Quiros, D 84, 103513 (2011); M. Carena, N. R. Shaha, and C. E. M. Phys. Rev. D 78, 123528 (2008). Wagner, Phys. Rev. D 85, 036003 (2012); A. Ahriche and S. [30] D. Comelli and J. R. Espinosa, Phys. Rev. D 55, 6253 Nasri, Phys. Rev. D 85, 093007 (2012); D. Borah and J. M. (1997). Cline, Phys. Rev. D 86, 055001 (2012). [31] S. W. Ham and S. K. Oh, Phys. Rev. D 70, 093007 (2004). [11] V. Q. Phong, H. N. Long, V. T. Van, and N. C. Thanh, Phys. [32] C. Grojean, G. Servant, and J. D. Wells, Phys. Rev. D 71, Rev. D 90, 085019 (2014); J. S. Borges and R. O. Ramos, 036001 (2005). Eur. Phys. J. C 76, 344 (2016). [33] C. Delaunay, C. Grojean, and J. D. Wells, J. High Energy [12] J. R. Espinosa, T. Konstandin, and F. Riva, Nucl. Phys. Phys. 04 (2008) 029. B854, 592 (2012). [34] X. Gan, A. J. Long, and L.-T. Wang, Phys. Rev. D 96, [13] S. Kanemura, E. Senaha, T. Shindou, and T. Yamada, J. 115018 (2017). High Energy Phys. 05 (2013) 066. [35] A. Kusenko, L. Pearce, and L. Yang, Phys. Rev. Lett. 114, [14] D. J. H. Chung and A. J. Long, Phys. Rev. D 81, 123531 061302 (2015). (2010). [36] B. Grzadkowski, M.Iskrzyski, M. Misiak, and J. Rosiek, [15] G. Barenboim and N. Rius, Phys. Rev. D 58, 065010 (1998). J. High Energy Phys. 10 (2010) 85. [16] S. Profumo, M. J. Ramsey-Musolf, and G. Shaughnessy, [37] K. Hagiwara, S. Ishihara, R. Szalapski, and D. Zeppenfeld, J. High Energy Phys. 08 (2007) 010; S. Profumo, M. J. Phys. Rev. D 48, 2182 (1993). Ramsey-Musolf, C. L. Wainwright, and P. Winslow, Phys. [38] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936) Rev. D 91, 035018 (2015); D. Curtin, P. Meade, and C.-T. (English translation); J. Schwinger, Phys. Rev. 82, 664 Yu, J. High Energy Phys. 11 (2014) 127; M. Jiang, L. Bian, (1951). W. Huang, and J. Shu, Phys. Rev. D 93, 065032 (2016). [39] J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127, [17] M. Carena, G. Nardini, M. Quiros, and C. E. M. Wagner, 965 (1962); G. Jona-Lasinio, Nuovo Cimento 34, 1790 Nucl. Phys. B812, 243 (2009); A. Katz, M. Perelstein, M. J. (1964). Ramsey-Musolf, and P. Winslow, Phys. Rev. D 92, 095019 [40] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973). (2015). [41] R. Jackiw, Phys. Rev. D 9, 1686 (1974). [18] H. H. Patel and M. J. Ramsey-Musolf, Phys. Rev. D 88, [42] J. Iliopoulos, C. Itzykson, and A. Martin, Rev. Mod. Phys. 035013 (2013); N. Blinov, J. Kozaczuk, D. E. Morrissey, 47, 165 (1975). and C. Tamarit, Phys. Rev. D 92, 035012 (2015); S. Inoue, [43] M. E. Peskin and D. V Schroeder, An Introduction to G. Ovanesyan, and M. J. Ramsey-Musolf, Phys. Rev. D 93, Quantum Field Theory (Addison-Wesley publishing, 015013 (2016). New York, 1995). [19] H. H. Patel and M. J. Ramsey-Musolf, J. High Energy Phys. [44] V. Mukhanov, Physical Foundations of Cosmology 07 (2011) 029; G. W. Anderson and L. J. Hall, Phys. Rev. D (Cambridge University Press, Cambridge, United Kingdom, 45, 2685 (1992). 2005).

116010-16 SPHALERON IN THE FIRST-ORDER ELECTROWEAK PHASE … PHYS. REV. D 101, 116010 (2020)

[45] G. W. Anderson and L. J. Hall, Phys. Rev. D 45, 2685 [48] S. Braibant, Y. Brihaye, and J. Kunz, Int. J. Mod. Phys. A (1992). 08, 5563 (1993). [46] F. R. Klinkhamer and N. S. Manton, Phys. Rev. D 30, 2212 [49] M. Chala, C. Krause, and G. Nardini, J. High Energy Phys. (1984). 07 (2018) 062. [47] K. Funakubo and E. Senaha, Phys. Rev. D 79, 115024 [50] C. L. Wainwright, Comput. Phys. Commun. 183, 2006 (2009). (2012).

116010-17