Gauge-independent T C and VEVs

The NFK identity [15, 16] tells us that^‡

∂Veff(ϕ, ξ)

∂ξ =−C(ϕ, ξ)∂Veff(ϕ, ξ)

∂ϕ , (2.33)

where C(ϕ, ξ) is a functional. In fact, T_Ccan be proved formally to be gauge-independent with the identity and Eq. (2.28)[4] as long as we are working on the full order of the effec-tive potential. However, in the practical calculation for T_C, one could only calculate the potential up to some order. For example, like in our previous standard method of deter-mining T_C includes only the one-loop order effect; thus, this causes an artificial violation of the NFK identify, even though the full order formalism is gauge-invariant. In order to regulate this issue, we have to keep tracking whether the identity is valid in each order of

¯

h. In the perturbation theory, in principle, we can expand V_effand C in the power of ¯h:

V_eff(ϕ) = V₀(ϕ) + ¯hV₁(ϕ, ξ) + ¯h²V₂(ϕ, ξ) +· · · , (2.34) C(ϕ, ξ) = c₀+ ¯hc₁(ϕ, ξ) + ¯h²c₂(ϕ, ξ) +· · · . (2.35)

By inserting them into Eq. (2.33), we get

¯

h∂V₁(ϕ, ξ)

∂ξ + ¯h²∂²V₂(ϕ, ξ)

∂ξ² +· · · = − c0

∂V₀

∂ϕ − ¯h (

c₁(ϕ, ξ)∂V₀(ϕ)

∂ϕ + c₀∂V1(ϕ, ξ)

∂ϕ )

+O(¯h²) +· · · ,

(2.36)

where we had expressed Eq. (2.33) in ¯h order. Since the tree-level potential is free of ξ-dependence, c0 = 0. AtO(¯h),

∂V₁(ϕ, ξ)

∂ξ = c₁(ϕ, ξ)∂V₀(ϕ)

∂ϕ . (2.37)

‡Even though our potential is a two dimensional (ϕ, ϕS) function, the procedure is straight forward: by taking the other dimensions as zero, one can get a pair of Eq. (2.33). Using the same method described in above, one can get similar results.

Eq. (2.37) shows that as long as we are working onO(¯h), the gauge dependence of V1(ϕ, ξ) is vanished at stationary point(s) of the tree-level potential. On the contrary, T_Cand VEVs determined by the standard method are evaluated in the tree-level plus one-loop potential minima; as a result, those results are gauge-dependent. Extending above to our model, the ξ dependence of V₁(ϕ, ϕ_S, ξ) disappears while we evaluate T_C at the tree-level minima:

(ϕ = v^tree = 246 GeV, ϕ_S = 0) and (ϕ = 0, ϕ_S = v^tree_S = √

µ²_S/λ_S). For the two-step EWPT, the gauge-independent T_C can be obtained from the following:

V₀(v^tree, 0) + V_CW(v^tree, 0) + V₁^T(v^tree, 0, T_C)

= V₀(0, v_S^tree) + V_CW(0, v_S^tree) + V₁^T(0, v_S^tree, T_C).

(2.38)

Note that as we are working onO(¯h), the thermal resummation, a two-loop order O(¯h²) effect, is not performed. Going beyondO(¯h) requires two-loop contributions which is out of scope of our current analysis. In the Appendix E, we give an example to explicitly show how the gauge dependence disappears when the potential is evaluated at the tree-level minima.

The minima, v_C and v_SC, are inherited gauge dependence which can easily be un-derstood from the NFK identity: in Eq. (2.33), the field value minimizing the effective potential is gauge-dependent. To get gauge invariant VEVs in the PRM scheme, we have to utilize the HT potential, Eq. (2.32), by finding the minima at T_C, i.e., find the minima of V^HT(ϕ, ϕ_S, T_C).

Chapter 3

Numerical Analysis

In our model, mS, λS and λHS are the free parameters. We take mS = mH/2 which is within DM experiment bound and phenomenology (see Sec. ?? for the discussion), and choose λ_S = λ^min_S + 0.1, see Eq. (2.8); thus only λ_HS is varied in the analyses. Aiming to investigate the gauge dependence, a range of λ_HS is selected where the tree-level po-tential is relatively minor compared with gauge loop contributions to the popo-tential barrier.

Furthermore, this particular range meets the conditions of the two-step strong first-order EWPT (v_C/T_C ≃ 1 and several inequalities in Sec. 2.3.1). For clarity, our input param-eters are listed in Table 3.1, and we summarize the settings of the each scheme in Table 3.2.

In Sec. 3.1, we list a detail procedure of finding T_Cand v_C; in addition, their precisions in the analyses are also shown. Our analyses can be divided into two parts: Sec. 3.2 focuses on the effect of gauge and the NG boson channels in the OS-like scheme, Sec. 3.3 demonstrates how the scheme dependence influences the results.

Parameter

m_H v₀ m_W m_Z m_t Value [GeV] 125 246 80.4 91.1 173.2

Table 3.1: The input parameters in all schemes. Note that, besides the MS scheme (see Sec. 2.2), µ_H, µ_S, and λ_H satisfy the tree-level relations: µ²_H = m²_H/2, µ²_S = −m²S + λ_HS/2v₀² and λ_H = m²_H/2v₀².

Scheme

OS-like MS HT PRM

Tadpole Condition Tree 1-loop Tree Tree

NG resummed √

- -

-Thermal resummed √ √

-

-ξ dependence √ √

-

-Table 3.2: Summary of the scheme settings.

3.1 Critical Temperature and Critical VEV

Two s of method of finding T_C are used; For the PRM scheme, T_C is found by Eq. 2.38.

For the OS-like, the HT and the PRM schemes, T_C is searched by the bisection method whose details of procedure are listed in the following^*:

1. choose two initialized temperatures, T_max(e.g., 200 GeV) and T_min(100 GeV). To use the bisection method, one also needs a middle value, T_mid = (T_max+ T_min)/2 (150 GeV),

2. calculate the energy difference, ∆E, between the potential energy of minima in singlet’s, E(ϕ^min), and doublet’s, E(ϕ^min_S ), directions^†at T_mid, i.e.,

∆E(T_mid) = E(ϕ^min, T_mid)− E(ϕ^minS , T_mid), (3.1)

3. calculate ∆E at T_min, that is

∆E(T_min) = E(ϕ^min, T_min)− E(ϕ^minS , T_min), (3.2)

4. if ∆E(T_mid)× ∆E(Tmin) < 0, we can redefine T_min → Tmin, and T_max → Tmid, otherwise T_min→ Tmidand T_max→ Tmax,

5. take the new Tmidand Tminback to the step 2. and go though the procedure again.

The above is calculated iteratively until

*We useMathematica to analyse our model.

†We use theFindminimum, the optional methods InteriorPoint and PrincipleAxis are chose.

• ∆E(Tmid) is less than 10³GeV,

• and the absolute temperature difference (|Tmid− Tmin|) is < 5 × 10⁻³ GeV,

• or the maximum calculation count (20) is reached.

Then we regard T_midas T_C; v_Care searched in this temperature. For the PRM scheme, the gauge-invariant v_C is found by using Eq. (2.32) at T_C which is obtained from Eq. (2.38).

In the Fig. 3.1, we show an example of the contour plots of the HT potential in the plane of⟨ϕ⟩ and ⟨ϕS⟩ for temperatures are high, above, equal and below TC; zero temperature one is also included. All the parameters are chose to be m_S = m_H/2 and λ_HS = 0.4.

3.2 (non)Thermal Gauge and NG Boson Contribution

To numerically quantify the effect of gauge and NG boson contributions, we use several approximations by turning off specific channel to study the EWPT, see Fig. 3.2. “(T)GB off”, depict by the (red) green (dot-dashed) dashed line, is denoted the computation without (thermal) non-thermal gauge boson contributions; “NG off”, depicted by the yellow dotted line, stands for NG boson contributions is omitted. “full”, depicted by the blue solid line, includes all the contributions. The left and right panels show TC and vC/T_C as functions of λ_HS, respectively.

Our results indicate that, in the range of 0.2 < λ_HS < 0.4, the thermal gauge loop have a∼ 11% effect on TC and∼ 13% on vC/T_C. As mentioned before, the result is not precisely equivalent to the effect of ξ dependence. However, it demonstrates how gauge boson contributions affect the EWPT; thus, the gauge artifact shoud have large impact on the EWPT as well. Furthermore, [24] found that even though the percentage of difference is small in TC as changing different ξ, it cannot guarantee that the dependence on gravitational waves generated from the first-order of EWPT is insignificant. In fact, by varying ξ from 0 to 5, the gravitational wave spectrum in a U(1)_B−Lmodel can change by one order of magnitude [24]. Above result indicates the necessity of quantification of the ξ dependence by using general R_ξgauge which should be studied and noticed in the future.

-300 -200 -100 0 100 200 300

Figure 3.1: The contour plots of the HT effective potential in the plane of⟨ϕ⟩ and ⟨ϕS⟩ at T = 300 GeV (Upper Left), T = T_C + 10 GeV (Upper Right), T = T_C = 75.1 GeV (Middle Left), T = T_C − 10 GeV (Middle Right) and T = 0 GeV (Lower). The parameters are, m_S = m_H/2 and λ_HS = 0.4. Note that all the plot legends are in the unit of 10⁷GeV.

In the NG off case, their effects are expected to be minor because that they are the two-loop level effects. However, when λ_HS . 0.21 where the thermal gauge loop dom-inates over the barrier compared with the tree-level contributions, NG boson effects are pronounced. To be noted that, in this region, the global minimum at T_Cis no longer located on the doublet Higgs; instead, it is mixed with the doublet and the singlet contributions;

this causes the downward curve of v_C/T_C. The (0.03− 2.3)% difference is found in TC,

Table 3.3: The percentage of the effects on the EWPT by turning off specific channel.

full GB off TGB off NG off

0.20 0.25 0.30 0.35 0.40

60

0.20 0.25 0.30 0.35 0.40

0.0

Figure 3.2: The effect of thermal (TGB off, red dot-dashed), non-thermal gauge (GB off, green dashed) and NG bosons (NG off, yellow dotted) contributions on T_C (Left panel) and v_C/T_C (Right panel) in the OS-like scheme. The blue solid line includes all the con-tributions.