Top Quark

As only the top quark contribution is significant for our analyses, its field-dependent mass is

m²_t(ϕ) = y²_t

2ϕ². (D.17)

Appendix E

An Illustrative Application PRM

Here we present an explicit example to demonstrate how the gauge dependence disappears in the PRM scheme. Consider a SU_L(2)× U(1) theory, the field is written as

H = 1

The generators^*of those groups are

T¹ = 1

*We use the real representation, i.e.,−i is factored out. That is, T = −iT , where T is what we adopt.

Note that{T¹, T², T³} are generators of isospin, and T⁴is for hypercharge. The Euclidean

The field can be separated by the classical fields (ϕc) and quantum fields, h(x):^†

Φ_i(x) = ϕ_{c i}+ h_i(x), (E.4)

where ϕ_c = (0, 0, ϕ, 0), and h = (h₁, h₂, h₃, h₄). The Higgs field is the third component of h. The location of extrema of the Higgs potential at the tree level are

ϕ⁽¹⁾₀ = 0, ϕ⁽²⁾₀ =±√

µ²/λ =± 246 GeV. (E.5)

Let’s expand Eq. (E.2) in terms of ϕ, arranging it in the power of the quantum field (h_i) and the gauge fields (W_µ). Then the Euclidean Lagrangian will be

LE =

Note that (1) since ϕ_c is independent of spacetime, its derivative is zero; (2) those terms with cubic and quadratic in ϕ_c and W_µ^a are omitted here; (3) the gauge field strength is

†This is called the background field method, a classical field are defined by: Euclidean action is mini-mized when the field value equals to the classical field. A quantum field is the field that fluctuates around the minimum.

cast by the following,

where we have integrated by parts and redefined µ and ν due to their symmetry. Eq. (E.6) is equivalent to a more compact form:

V₀(ϕ_c) + ϕ_i∂V₀(Φ) The explicit form of above can be understood by expanding the component in the potential:

V₀(Φ) =−1

and thus the derivative can be understood by the following: if i̸= 3,

∂V₀

+ purely quantum field terms, (E.12)

∂²V0

∂h_i∂h_j 

hc = (−µ²+ λϕ²)δ_ij, (E.13)

and if i = 3,

Notice that ab indices of the bracket represent the gauge fields’ indices. The mixing term,

∂^µϕ_iW_µ^a(gT^aϕ_c)_i, in Eq. (E.7) can be removed by imposing the gauge fixing condition,

F^a= ∂^µW_µ^a− ξϕi(gT^aϕ_c)_i = 0, (E.19)

with the gauge fixing term

where the last term is called gauge-fixing scalar boson mass matrix:

m²_A(ϕ_c)_ij = (gT^aϕ_c)^T_i × (ϕi(gT^aϕ_c)_j

note that the ij indices of the bracket represent the scalar fields’ indices. We also add in the ghost compensating terms:

Lgh = η^†a(

−∂²δ^ab− ξm²A(ϕ_c)^ab)

η^b+ gf^abc(∂^µη^†a)− ξ(gT^aϕ_c)_iη^†aη^b(gT^bϕ_c)_j.

(E.22)

The full Lagrangian is then

where high order terms are omitted for simplicity. The results of loop-level potential calculation give us that

V_eff(ϕ) = V₀(ϕ) (E.24)

where· · · are field independent terms. A pre-factor, d − 1, is from the dimension regular-ization (the conventional choice is d = 4− 2ϵ). This result shows that Eq. (E.25) comes from the scalar loop, Eq. (E.26) comes form the gauge loop; (E.27) is originated from the ghost loop. Note that the gauge dependence is cancelled out, if the scalar loop contribution is arranged into

Tr ln(p²− Mij² − ξm²A(ϕ)_ij)−→ Tr ln(p²− Mij²) + Tr ln(p²− ξm²A(ϕ)_ij); (E.28)

As a consequence, the gauge dependence will be eliminated at the one-loop level. The main idea of the PRM scheme is to achieve the above arrangement. Let’s first consider a matrix M whose matrix elements A, B, etc., are also matrices:

M =

Since M can be arranged into

M =



A 0 C I







I A⁻¹B 0 D− CA⁻¹B



 , (E.29)

and the property of the determinant shows that det G = det(T F ) = det(T ) det(F ), its determinant is then

det M = det A det(D− CA⁻¹B). (E.30) If B, C are zeros, det M = det A det D. As we know that ln det M = Tr ln M ,

ln det M = ln det A + ln det D, Tr ln M = Tr ln A + Tr ln D.

(E.31)

Back to our case, the matrix in Eq. (E.28) logarithm is

S_ij = p²_ij − M²(ϕ)_ij− ξm²A(ϕ)_ij. (E.32)

If S_ij is a block-diagonalized matrix which is crucial since in the most of the cases it is not; then M_ab² and m²_{A cd} are simultaneously diagonalizable but their eigenvalues live in distinct subspaces (i.e., their eigenvalues are not mixed), then we have^‡



p²I_ab− Mab² 0 0 p²_cd− ξm²A cd



 . (E.33)

As a result, combining Eq. (E.31), we finally have the separation:

Tr ln S = Tr ln(p² − M_ab²) + Tr ln(p²− ξm²_{A cd}). (E.34)

In this particular circumstance, we can eliminate the gauge dependence at the one-loop level. In the next section, we see how the gauge-invariant T_C can be got from the above formalism.

‡In the Sec. E.1, the upper left matrix is 1× 1, and the lower right matrix is 3 × 3

E.1 Gauge-invariant T

_C

In this theory, similar to the condition in our model Eq. (2.38), the PRM scheme finds TC

when the condition,

V₀(ϕ⁽¹⁾₀ ) + ¯hV₁(ϕ⁽¹⁾₀ , T_C, ξ) = V₀(ϕ⁽²⁾₀ ) + ¯hV₁(ϕ⁽²⁾₀ , T_C, ξ), (E.35)

is satisfied; the minima are defined at Eq. (E.5). If ξ terms which mentioned in the previous section is cancelled, this particular T_C is gauge independent. In Eq. (E.35), the left hand side is gauge invariant which can be easily recognized: all the gauge dependent terms Eq. (E.18) and Eq. (E.21) in Eq. (E.25), Eq. (E.26) and Eq. (E.27) are all proportional to the classical background whose ϕ⁽¹⁾₀ = 0. The right hand side takes the EW-breaking minimum (ϕ⁽²⁾₀ = √

µ²/λ = 246 GeV). Therefore, the NG boson masses are equal to zero in this scheme, the scalar mass matrix Eq. (E.17) then becomes

M²(ϕ⁽²⁾₀ )_ij = diag(0, 0, 2µ², 0). (E.36)

The gauge-fixing scalar boson mass matrix is then

m²_A(ϕ⁽²⁾₀ )_ij = µ²

4λdiag(g₂², g₂², 0, g₁²+ g₂²). (E.37)

These two matrices actually can be arranged into the block-diagonalized form like Eq. (E.33), i.e., M²(ϕ⁽²⁾₀ )_ij and m²_A(ϕ⁽²⁾₀ )_ij are simultaneously diagonalizable; their eigenvalues live in different subspace.^§ Therefore, the separation like Eq. (E.34) can be achieve. To see explicitly how Tr ln(p² − ξm²A(h⁽²⁾₀ )_ij) eliminates Tr ln(p² − ξm²A(h⁽²⁾₀ )^ab), we need to

§Note that this is not always valid, only when one evaluates the minima at the tree level.

diagonalize m²_A(ϕc)^ab, Eq. (E.18), through the rotation matrix

R^ab =









 1

1

cos θ_W sin θ_W

− sin θW cos θ_W











, tan θ_W = g₂

g₁. (E.38)

After rotation, we have

R^abm²_A(ϕ_c)^bc(R^T)^cd = 1 4









 g₂²

g₂² 0

g₁²+ g²₂











ϕ². (E.39)

This is exactly the same as the gauge-fixing scalar boson mass matrix. Therefore, these two terms are cancelled out eventually. Finally, we show that T_C obtained from Eq. (E.35) is a gauge-independent result.

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