EWPT is revisited with the scalar singlet-extended SM by using several calculation schemes.
The effect of thermal, non-thermal gauge channels and NG resummation are studied. The parameters space we investigate is mS = mH/2 and 0.2≤ λHS ≤ 0.4.
In the OS-like scheme, the occurrence of the IR divergent in the NG boson channels demonstrates that the NG resummation is necessary; even though the NG resmmmation is the two-loop effect which is expected to be minor. We find that the NG resummation has a (0.03− 2.3)% effect on TC and (3.0− 16.7)% on vC/TC. When the non-thermal gauge channels are turned off, small numerical impacts on TC(0.16−0.18%) and vC/TC (0.29− 0.47%) are observed. The effect of thermal gauge channels is relatively significant even when the tree barrier is present, (11.4− 11.8)% on TC and (11.7− 16.8)% on vC/TC are detected. The above results have motivated us to investigate and quantify the ξ dependence by using the general Rξ in a future work.
We find that both TC and vC/TC in the OS-like and the MS schemes have nice agree-ment on each other; the differences are within the scale uncertainty. For TC, their dif-ference is (0.4− 3.2)% and for vC/TC is (0.03− 15)%. In the analysis between gauge-dependent and -ingauge-dependent schemes, we find that i) the HT scheme is over-simplified and TC is largely underestimated, and ii) we find that the PRM scheme is qualitatively con-sistent to the OS-like and the MS schemes. However, the large theoretical uncertainties caused by renormalization scale demonstrates the higher order corrections are needed.
Appendix A
Generating Functional of 1(not 1)-PI
The effective action can generate 1PI correlation functions. To see this, we first begin by the second derivative of the generating functional for the connected Green’s functions (W [J ]):
where we have omitted the position variable in the parentheses. Above can be simplified by using Eq. (1.5) notation (the subscript, J , is omitted for simplicity),
δ2W [J ]
δJ (y)δJ (z) =⟨0|ϕ(y)ϕ(z)|0⟩ − ⟨0|ϕ(y)|0⟩⟨0|ϕ(z)|0⟩. (A.2)
The physical meaning of the above can be shown diagrammatically, the first term in the right hand side includes both*
x y
, (A.3)
*Diagrams are generated by [32].
x + y
, (A.4)
where every blob contains sum of connect diagrams. For the second term of Eq. (A.2), it contains only diagrams like Eq. (A.4). Thus, W [J ] generates diagrams like Eq. (A.3) alone; furthermore, the higher derivatives of W [J ] will still generate the connect diagrams as desired. The effective action generates the inverse of the same two-point diagrams as W [J ]. To see this, from Eq. (1.9), we know that
Furthermore, substituting ϕcwith Eq. (1.5), we have
− δ(x − y) =
one can easily observe that the second derivative of the effective action is actually the inverse of the two point function. At this point, one can already notice that the generating function of the effective action; however, to demonstrate the ability of generating 1PI, one has to go to the 3rd or higher derivative of the effective action. Before showing the main difference of Γ[ϕc] and W [J ], we first consider
to evaluate the derivative of the parentheses we use
∂
∂αM−1(α) =−M−1∂M(α)
∂α M−1. (A.9)
Consequently, we have δ3W [z]
δJ (x)δJ (y)δJ (z) =−i
∫
d4w D(z, w)
∫
d4ud4v
( δ2Γ δϕc(x)ϕc(u)
)−1
δ3Γ
δϕc(u)δϕc(v)δϕc(w)
( δ2Γ δϕc(v)ϕc(y)
)−1 ,
= i
∫
d4wd4ud4v D(z, w)D(x, u) δ3Γ
δϕc(u)δϕc(v)δϕc(w)D(v, y), (A.10)
where in the last step, Eq. (A.7) is used. The equation can be understood by the digram again:
z x
y
= z w v y
u x
, (A.11)
where the left hand side digram equals to the 3rd derivative of W [J ]; the two-point func-tions connect the internal posifunc-tions u, v and w, which will be integrated out later, to the external points (x, y and z). In the right hand side, the crossed dot which connects the the internal positions represents the 3rd derivative of the effective action; since the exter-nal lines are amputated, one can observe that the crossed dot is actually a 1PI three-point function which is generated by the effective action. Additionally, one can show that for n≥ 3, the effective action is the generating functional for 1PI n-point Green’s function.
Appendix B
Effective Potential in One-Loop
To calculate the one-loop effective potential, recall the general form, Eq. (1.24), for con-venience,
V1(ϕc) =−i 2
∫ d4p
(2π)4Trln iG−1(p; ϕc). (B.1) We first consider a simple theory with a scalar field, where its Lagrangian is given by
L = 1
2(∂µϕ(x))2− 1
2m2ϕ2(x)− λ 4!ϕ4(x), and V0 = 1
2m2ϕ2(x) + λ
4!ϕ4(x).
(B.2)
The propagator of this theory is then
G(p; ϕc) = i
p2− m2(ϕc) + iϵ, (B.3)
where the definition of m(ϕc) can be found in Appendix D while λHS is turned off. Since the scalar field is absent of internal d.o.f., the trace in Eq. (B.1) is equal to 1; so that it becomes
V1(ϕc) =−i 2
∫ d4p
(2π)4ln (p2− m2(ϕc) + iϵ). (B.4)
In order to evaluate the integral, a Wick rotation is performed:
where we have suppressed the iϵ term and subtracted a constant coefficient, ln(−1). In the second equality, we have used the dimension regularization. In the D-dimension, λ has the dimension of 4− D ≡ ε; however, instead of using a non-integral dimensional coupling constant, we introduce a parameter (µ), which later is recognized as a renormalization scale and it has mass dimension, and trade λ→ λµε. To unfold the logarithm, one has to calculate the derivative with respect to m2,
µε∂V1(ϕc)
where the identity [33],
∫ dDk has been used. Integrating with respect to m2,
µεV1(ϕc) = 1
where we have used the Γ function’s property: Γ(1 + z) = zΓ(z). Taylor expansions have also been performed; thus after utilizing the MS renormalization scheme (we also present DR scheme for the completeness. In this case, these two schemes give the same results.) and taking ε→ 0, we have
V1scalar(ϕc) = m4(ϕc) 64π2
(
lnm2(ϕc)
¯
µ2 −3 2
)
, (MS-scheme = DR-scheme), (B.9)
where ¯µ2 = 4πe−γEµ2. Notice that in the following, we adopt procedures that i) the dimensional regularization scheme is followed by the MS renormlization scheme; ii) the dimensional reduction is renormlized by the DR scheme. In the thesis, we use the former method.
For fermion, the relevant part of Lagrangian is
L(ϕ, ψ) ∼ ¯ψ(
∂/− mf(ϕc))
ψ, (B.10)
and its propagator is
Gf = i
p2 − m2f(ϕc) + iϵ. (B.11) Note that only diagrams with even number of vertexes have contributions, since the trace of odd number gamma matrix(s) is zero. The internal d.o.f. of fermion is the spinor space whose trace actually is depended on the regularization schemes,
Tr(1) =
2D/2 = 2ε/2−2, dimensional regularization, 4, dimensional reduction.
(B.12)
Accordingly, the one-loop potential for fermion case is then
V1(ϕc) = i 2
∫ d4p
(2π)4Tr(1)Trln(p2− m2f(ϕc)), (B.13)
where the addition minus comes from the fermionic loop; the second trace is for the dif-ferent fermion species. By performing the similar procedure like above, we have
V1fermion(ϕc) =
Note that the factor, ln 2, is often omitted in the convention, as it is absorbed in ¯µ. If the perturbation is good enough, the small difference of ¯µ will not change the result signifi-cantly; we also mention how this issue affect TC in the Sec. 3.4.
Above is also applied to the boson channels. For boson case, gauge bosons related parts in Lagrangian are
ab, one can check Eq. (E.18). After taking the Landau gauge, the propagator is
Gg = i
where the first trace is for the different channels of the gauge bosons; the second trace is depended on regularization scheme:
Tr (∆µν) = gµν∆µν
= 3, dimension reduction.
(B.18)
Finally, the one-loop potential for gauge bosons are given by
V1boson(ϕc) =
3×M64πg4(ϕ2c)
( lnM
2 g
¯
µ2 − 56)
, (MS-scheme), 3×M64πg4(ϕ2c)
( lnM
2 g
¯
µ2 − 32)
, (DR-scheme).
(B.19)
Appendix C
Approximate Thermal Function
The thermal functions are infinite sums of modified Bessel functions of the second kind (K2) [22]:
IB,F(a2) =−a2
∑∞ n=1
(±1)2
n2 ℜK2(na), (C.1)
where the plus and minus signs are for the boson and fermion, respectively. We set the goal of the difference between approximate functions and the numerical integrations should be less than 10−6. To achieve this, we investigate the following:
C.1 Boson
In the region of a = m/T < 0.35, we approximate the thermal function (IB) with a high temperature expansion for boson (HTEB); in 0.35 ≤ a ≤ 9.0, a polynomial fitting function for boson (PFFB) is used; a low temperature approximation for boson (LTEB) is adopted for a > 9.0; when a∈ iℜ, we replace ℜK2 with the second kind of unmodified Bessel function (Y2) by an identity.