Aspects of the Higgs Pole Mass in Various Gauges

Vol. 70, No. 1, January 2020, pp. 19∼23 http://dx.doi.org/10.3938/NPSM.70.19

Aspects of the Higgs Pole Mass in Various Gauges

Chungku Kim

Department of Physics, Keimyung University, Daegu 42601, Korea (Received 11 October 2019 : accepted 3 December 2019)

We investigated the determination of the pole mass for the broken symmetry phase of the Abelian Higgs model in the Rξ, Landau and the unitary gauges. In the case of the Rξ and Landau gauges, we obtained the same finite renormalized two-point functions, which were scale independent. Es- pecially in the Landau gauge, the basic one-loop integral with zero mass B(0, 0) appearing in the renormalized two-point function which was originated from the ξ dependent masses, canceled out in the resulting pole mass. In the case of the unitary gauge, the problem of renormalization occurs in the two-point function, and if we take only the finite part of the p⁴term in the two point function, then we obtain the same result for the pole mass as in case of the Rξ and Landau gauges.

PACS numbers: 11.15.Bt, 12.38.Bx Keywords: Pole mass, Gauge

I. Introduction

In the calculation of the physical quantities of the quantized field theory, we need to fix a gauge and the re- sult should be independent of the choice of the gauge [1].

However, recently, it turned out that the decay width of the H− > γγ in the Rξ and unitary gauges do not coin- cide [2] and gauge independence of the physical quanti- ties are not valid in this example.

The pole mass plays an important role in the process where the characteristic scale is close to the mass shell [3] and in this paper, as an another example of deter- mining the physical quantities in different choices of the gauges, we will investigate the pole mass for the broken symmetry phase of the Abelian Higgs model in the Rξ, Landau and unitary gauges. Landau gauge was used in higher order loop calculation of the Higgs pole mass in order to simplify the calculation [4,5]. In case of the Rξ

gauge, by using the fact that the renormalization of the vacuum expectation value(VEV) is different from that of the Higgs field and using the ¯h-expansion [6], we obtain the Higgs pole mass in the broken symmetry phase of the Abelian Higgs model which is invariant under the renor- malization group(RG) function in the symmetric phase.

∗E-mail: [email protected]

II. Higgs Pole Mass in SSB Phase of the Abelian Higgs Model

1. Rξ gauge

The bare Euclidean Lagrangian density of the Abelian Higgs model in Rξ gauge [7] is given by

L =1

4FµνBFµνB+1

2(∂µΦ1B+ gBAµBΦ2B)² +1

2(∂µΦ2B− gBAµBΦ1B)²−1

2m²_B(Φ²_1B+ Φ²_2B) + 1

24λB(Φ²_1B+ Φ²_2B)²+ 1 2ξB

(∂µAµB− gBvBΦ2B)²,

where

FµνB = ∂µAνB− ∂υAµB. (1)

The bare fields, coupling constants and the VEV are re- lated to the renormalized fields, coupling constants and the VEV as λ

Φ1,2B =√

ZϕΦ1,2, AµB =√ ZAAµ, m²_B= Z_mm², λ_B= Z_λλ and v_B= Z_vv,

(2)

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

with the ¯h-expansion as

Z_m = 1 + ¯hδ_m⁽¹⁾+ ¯h²δ_m⁽²⁾+· · · , Zλ = µ^2ε(1 + ¯hδ⁽¹⁾_λ + ¯h²δ_λ⁽²⁾+· · · ), Zϕ = µ^−ϵ(1 + ¯hδ_ϕ⁽¹⁾+ ¯h²δ_ϕ⁽²⁾+· · · ), ZA = µ^−ϵ(1 + ¯hδ_A⁽¹⁾+ ¯h²δ_A⁽²⁾+· · · ), Zv = 1 + ¯hδ_v⁽¹⁾+ ¯h²δ_v⁽²⁾+· · · and

v = v0+ ¯hv1+ ¯h²v2+· · · . (3) The one-loop counterterms in D = 4− 2ε dimensional regularization scheme and corresponding renormaliza- tion group functions in the symmetric phase are well known:

δ⁽¹⁾_λ = κ ε(5

3λ− 6g²+ 18g⁴ λ), δ⁽¹⁾_g = κ

ε 1 3g², δ⁽¹⁾_m = κ

ε(2

3λ− 3g²) and δ⁽¹⁾_ϕ = κ

ε(3− ξ)g², (4)

and by using the RG functions in the MS scheme [8,9]

β_λ⁽¹⁾ = κ(10

3 λ²− 12λg²+ 36g⁴), β_g⁽¹⁾ = κg³

3 , β_m⁽¹⁾2 = κ(4

3λ− 6g²)m²,

γ_ϕ⁽¹⁾ = κ(3− ξ)g², (5)

where κ≡ _16π¹2. Since the l−loop Feynman diagram con- tains the factor ¯h^l, at the order ¯h^l-th order calculation of the ¯h-expansion, we should include both the l−loop Feynman diagram which contains the order ¯h⁰ parame- ters g, λ, m and v0and the k(≺ l)-loop Feynman diagram which contains the order ¯h^a parameters δ^(a)and vbsuch that a + b = l− k.

In the broken symmetry phase, we substitute ΦB+ v_B for Φ1B. Recently, it was shown that Zϕ̸= Zv in case of the Rξ gauge [10,11] and we should determine Zv from the vanishing tadpole condition in the broken symmetry phase. Then we obtain the Higgs part of the Lagrangian LH as

L_H=(−m²B

√Z_ϕ√

Z_vv +1 6λ_B√

Z_ϕ√

Z_v³v³)Φ + (−1

2m²_BZ_ϕ+1

4λ_BZ_ϕZ_vv²)Φ² +1

6λ_B√ Z_ϕ³√

Z_vvΦ³+ 1

24λ_BZ_ϕ²Φ⁴.

(6)

In order to determine the one-loop renormalization con- stant of the VEV δv⁽¹⁾, by using Φ term of Eq.(7), let us consider vanishing tadpole condition for one-loop.

− m²v₀(1 +1

2δ⁽¹⁾_ϕ +1

2δ⁽¹⁾_v )− m²v₀δ⁽¹⁾_m − m²v₁ +1

6λv³₀(1 + δ_λ⁽¹⁾+1

2δ_ϕ⁽¹⁾+3 2δ⁽¹⁾_v ) +1

2λv²₀v₁+1

2λv₀A(m²_H) + 3g²v₀A(m²_A) + 1

18λv0A(ξm²_A),

(7)

where

m²_H= 2m², m²_A= g²v₀², (8) and the one-loop function A(m²) and B(m²₁, m²₂) is given by [12]

A(m²) =

∫ d^Dq (2π)^D

1

q²+ m², (9) and

B(m²₁, m²₂) =

∫ d^Dq (2π)^D

1

(q²+ m²₁)((p + q)²+ m²₂). (10) Order ¯h⁰ term of Eq.(8) determines v²₀ as

v²₀=6m²

λ , (11)

and at order ¯h¹,¹_εdivergence can be removed by choosing

δ_v⁽¹⁾= κ

ε(3 + ξ)g², (12) which agrees with the previous results obtained by dif- ferent methods [13]. Then the remaining finite of order

¯

h terms of Eq.(8) determine the one-loop VEV as λv0v1=

[

−3

2λA(m²_H)− 9g²A(m²_A)−1

2λA(ξm²_A) ]

f in

, (13) where we use the notation [X]_{f in} meaning that finite part of X in which ¹_ε divergence is removed. Now let us turn to the one-loop (order ¯h) the two-point function of Abelian Higgs model in the SSB phase. By using Φ² term of Eq.(7), we obtain the unrenormalized Higgs two point function Π(p²)_{U R} as

Π(p²)U R= p²+ m²_H− m²(δ⁽¹⁾_m + δ_ϕ⁽¹⁾) +1

2λv₀²(δ_λ⁽¹⁾+ δ⁽¹⁾_ϕ + δ_v⁽¹⁾) + λv₀v₁+ Π_AA(p²) + ΠAΦ2(p²) + ΠΦ2Φ2(p²)

+ Π_HH(p²) + Π_A+ Π_Φ2+ Π_c+ Π_H,

(14)

where

ΠXY = 

X

Y , ΠX= 

X

. (15) By using the propagator for the gauge fields and the Goldstone boson Φ2 as

Aµ Aν =gµν−^pµ_p^p2^ν

p²+ m²_A + ξ^pµ_p^p2^ν

p²+ ξm²_A, (16) and

Φ2 Φ2 = 1

p²+ ξm², (17) and the well known one-loop integrals in D dimension, we can verify that the ¹_ε divergence vanishes with the counterterms given in Eqs.(5) and (13) and obtain the renormalized two-point function Π(p²)R as

Π(p²)_R = p²+ m²_H+ λv₀v₁

+



















 g²p⁴

2m²_A{B(ξm²A, ξm²_A)− B(m²A, m²_A)} +g²p²

m²_A {A(m²A)− A(ξm²A)}

− 2g²p²B(m²_A, m²_A) +λ

6A(ξm²_A)

−λ

6m²_HB(ξm²_A, ξm²_A)

− 6g²m²_AB(m²_A, m²_A) + 3g²A(m²_A) +λ

2A(m²_H)−3

2λm²_HB(m²_H, m²_H)





















f in

(18).

Note that there is no counterterm for the ¹_ε divergence of the p⁴ term and the p⁴ term becomes finite due to the fact that B(m²_A, m²_A) and B(ξm²_A, ξm²_A) have same ¹_ε divergence. The Higgs pole mass M_H² is de- termined as the pole of the renormalized one-particle- irreducible(1PI) Higgs two point function as

[Π(p²)R

]

p²=−MH²

= 0. (19)

Then, solving this equation iteratively by using Eqs.(19) and (20), we see that the ξ dependent one-loop inte- grals such as A(ξm²_A) and B(ξm²_A, ξm²_A) cancels out completely and obtain the gauge independent Higgs pole

mass as

M_H² = m²_H+ [

{−λ

6m²_H+2

3λm²_A− 6g²m²_A}B(m²A, m²_A)

− (λ

3 + 6g²)A(m²_A)− λA(m²H)

−3

2λm²_HB(m²_H, m²_H) ]

f in

.

(20) Moreover, by using

µ ∂

∂µA(m²) =−2m² and µ ∂

∂µB(m²₁, m²₂) = 2, (21) we can see that

µ d

dµM_H² = 0, (22)

which means that the Higgs pole mass is scale indepen- dent up to one-loop order.

2. Landau gauge

Landau gauge is the ξ→ 0 limit of the Rξ gauge and the propagator for the gauge fields and the Goldstone boson Φ2becomes

Aµ Aν =g_µν−^pµ_p^p2^ν

p²+ m²_A , (23) and

Φ2 Φ2 = 1

p², (24)

and the ghost field decouple. Then, by following same steps as in Rξ gauge we obtain the

λv₀v₁= [

−3

2λA(m²_H)− 9g²A(m²_A) ]

f in

, (25)

and the unrenormalized Higgs two point function Π(p²)U R as

Π(p²)_{U R} =p²+ m²_H− m²(δ⁽¹⁾_m + δ_ϕ⁽¹⁾) +1

2λv²₀(δ_λ⁽¹⁾+ 2δ_ϕ⁽¹⁾) + λv0v1+ ΠAA(p²) + ΠAΦ₂(p²) + ΠΦ₂Φ₂(p²)

+ ΠHH(p²) + ΠA+ ΠH.

(26)

In Landau gauge, ΠΦ₂ = 0 and by using the counterterms given in Eqs.(4) and (13) with ξ = 0, we obtain the renormalized two point function Π(p²)R as

Π(p²)R = p²+ m²_H+ +λv0v1

+

















 g²p⁴

2m²_A{B(0, 0) − B(m²A, m²_A)} +g²p²

m²_A A(m²_A)− 2g²p²B(m²_A, m²_A)

−λ

6m²_HB(0, 0)− 6g²m²_AB(m²_A, m²_A) + 3g²A(m²_A) +λ

2A(m²_H)

−3

2λm²_HB(m²_H, m²_H)



















f in

(27)

. Although in Landau gauge, the renormalized two point function becomes different from that of the Rξ gauge given in Eq.(19), we can check that the pole mass deter- mined by Eq.(20) becomes exactly the same as that of the Rξ gauge given in Eq.(21).

3. Unitary gauge

In unitary gauge, which is equivalent to ξ → ∞ limit of the Rξ gauge [14], only the gauge field and the Higgs field contribute to the unrenormalized Higgs two point function Π(p²)U R as

Π(p²)_{U R}=p²+ m²_H− m²(δ⁽¹⁾_m + δ_ϕ⁽¹⁾) +1

2λv₀²(δ⁽¹⁾_λ + 2δ⁽¹⁾_ϕ ) + λv₀v₁+ Π_AA(p²) + Π_HH(p²) + Π_A+ Π_H,

(28) with the propagator for the gauge field

Aµ Aν =

g_µν+^p_m^µp₂^ν

A

p²+ m²_A . (29) By using the one-loop counterterms given in Eqs.(5) and (13) and the one-loop VEV given in Eq.(26) which is ξ independent, we obtain

Π(p²)U R = p²+ m²_H+ λv0v1− g²p⁴

2m²_AB(m²_A, m²_A)

+







 g²p²

m²_A A(m²_A)− 2g²p²B(m²_A, m²_A)

− 6g²m²_AB(m²_A, m²_A) + 3g²A(m²_A) +λ

2A(m²_H)−3

2λm²_HB(m²_H, m²_H)









f in

(30).

It is known that the unitary gauge has a problem in the renormalization [14] and as we can see in Eq.(31), the p⁴ term in Π(p²)_{U R} is non-renormalizable contrary to the Rξ and Landau gauge where the ¹_ε divergence of the p⁴ term was cancelled by the contribution from the Goldstone boson diagrams. If we ignore this problem and take only finite part of p⁴term in Π(p²)U R, then we can obtain same results for the pole mass as in Rξ and Landau gauges.

III. Discussions and Conclusions

In this paper, as an another example of the physical quantities in different gauges, we have investigated the determination of the pole mass for the broken symme- try phase of the Abelian Higgs model in the Rξ, Landau and unitary gauge. In case of the Rξ and Landau gauge, we obtained the finite renormalized two-point function and have obtained same pole mass which was scale inde- pendent. Especially in Landau gauge, the basic one-loop integral with zero mass B(0, 0) in the renormalized two- point function canceled out in the resulting pole mass.

The origin of this zero mass in B(0, 0) is ξ dependent masses appearing either from the Goldstone boson or longitudinal part of the gauge boson whose mass be- comes zero in the Landau gauge which is the ξ→ 0 limit of the Rξ gauge. Since the pole mass should be gauge independent, this zero mass originating from the ξ de- pendent masses should not appear in the pole mass and this fact should be true in the higher order calculation of the pole mass where the Landau gauge was adopted for calculational simplicity. Then the zero mass in the ba- sic integral of l−loop calculation should appear only in diagrams where the internal lines have zero masses not originating from ξ → 0 limit of ξm². This means that only the basic integrals including the photon internal lines can have zero mass in case of the standard model.

Finally, in case of the unitary gauge, the problem of the renormalization occurs in the two-point function and if we take only finite part of the p⁴ term in the two-point function, then we obtain same result for the pole mass as in case of Rξand Landau gauges.

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