Gauge independent effective potential and the Higgs boson mass bound

Gauge independent effective potential and the Higgs boson mass bound

Guey-Lin Lin and Tzuu-Kang Chyi

Institute of Physics, National Chiao-Tung University, Hsinchu, Taiwan, Republic of China ~Received 2 November 1998; published 26 May 1999!

We introduce the Vilkovisky-DeWitt formalism for deriving the lower bound of the Higgs boson mass. We illustrate the formalism with a simplified version of the standard electroweak model, where all charged boson fields as well as the bottom-quark field are disregarded. The effective potential obtained in this approach is gauge independent. We derive from the effective potential the mass bound of the Higgs boson. The result is compared to its counterpart obtained from the ordinary effective potential.@S0556-2821~99!06211-6# PACS number~s!: 11.15.Ex, 12.15.Ji, 14.80.Bn

I. INTRODUCTION

The effective potentials in quantum field theories are off-shell quantities. Therefore, in gauge field theories, effective potentials are gauge-dependent as pointed out by Jackiw in the early 1970s @1#. This property caused concerns on the physical significance of effective potentials. In a work by Dolan and Jackiw@2#, the effective potential of scalar QED was calculated in a set of R_j gauges. It was concluded that only the limiting unitary gauge gives a sensible result on the symmetry-breaking behavior of the theory. This difficulty was partially resolved by the work of Nielsen @3#. In his paper, Nielsen derived the following identity governing the behavior of effective potential in a gauge field theory:

S

j_]j] 1C~f,j! ]

]f

D

V~f,j!50, ~1! wherejis the gauge-fixing parameter,fis the order param-eter of the effective potential, and C(f,j) is the Green’s function for certain composite operators containing a ghost field. The above identity implies that, for different j, the local extrema of V are located along the same character-istic curve on the (f,j) plane, which satisfies dj 5df/@C(f,j)/j#. Hence covariant gauges with different j are equally good for computing V. On the other hand, a choice of the multi-parameter gauge Lg f52(1/2j)(]mAm 1sf11rf2)2 @2#, with f1,2 the components of the scalar field, would break the homogeneity of Eq.~1! @3#. Therefore an effective potential calculated in such a gauge does not have a physical significance.

Recently, it was pointed out@4# that the Higgs boson mass bound, which one derives from the effective potential, is gauge dependent. The gauge dependence resides in the ex-pression for the one-loop effective potential. Boyanovsky, Loinaz and Willey proposed a resolution@5# to the problem, which is based upon the physical effective potential con-structed as the expectation value of the Hamiltonian in physi-cal states@6#. They computed the physical effective potential of an Abelian Higgs model with an axial vector coupling of the gauge fields to the fermions. A gauge-independent lower bound for the Higgs boson mass is then determined from the effective potential. We note that their approach requires the identification of first-class constraints of the model and a projection to the physical states. Such a procedure is not

manifestly Lorentz covariant. Consequently we expect that it is highly non-trivial to apply their approach to the standard model~SM!. In our work, we shall introduce the Vilkovisky-DeWitt formalism @7,8# for constructing a gauge-independent effective potential, and therefore obtain a gauge-independent lower bound for the Higgs boson mass.

In the Vilkovisky-DeWitt formalism, fields are treated as vectors in the configuration space, and the affine connection of the configuration space is identified to facilitate the con-struction of an invariant effective action. Since this proce-dure is completely Lorentz covariant, the computations for the effective potential and the effective action are straight-forward. We shall perform a calculation with respect to a toy model @9# which disregards all charged boson fields in the SM. It is easy to generalize our calculations to the full SM case. In fact, the applicability of Vilkovisky-DeWitt formal-ism to non-Abelian gauge theories has been extensively demonstrated in the literature@10#.

The outline of this paper is as follows. In Sec. II, we briefly review the Vilkovisky-DeWitt formalism using the scalar QED as an example. We shall illustrate that the effec-tive action of Vilkovisky and DeWitt is equivalent to the ordinary effective action constructed in the Landau-DeWitt gauge@11#. In Sec. III, we calculate the effective potential of the simplified standard model, and the relevant renormaliza-tion constants of the theory using the Landau-DeWitt gauge. The effective potential is then improved by the renormaliza-tion group analysis. In Sec. IV, the Higgs boson mass bound is derived and compared to that given by the ordinary effec-tive potential in the Landau gauge. We conclude in Sec. V, with some technical details discussed in the Appendix.

II. VILKOVISKY-DEWITT EFFECTIVE ACTION OF SCALAR QED

The formulation of Vilkovisky and DeWitt is motivated by the parametrization dependence of the ordinary effective action, which can be written generically as @12#

expi \ G@F#5exp i \

S

W@ j#1Fi dG dFi

D

5

E

@Df#exp_\i

S

S@f#2~fi2Fi!dG dFi

D

, ~2! where S@f# is the classical action, and Fi denote the back-ground fields. The dependence on the parametrization arises

because the quantum fluctuation hi[(fi2Fi) is not a vector in the field configuration space, hence the product hi

•dG/dFi _{is not a scalar under a reparametrization of}

fields. The remedy to this problem is to replace hi with a two-point function si(F,f) @7,8,13# which, at the point F, is tangent to the geodesic connectingF andf. The precise form ofsi(F,f) depends on the connection of the configu-ration space,G_jki . It is easy to show that@12#

si_~F,f!5hi₂1

2Gj k

i _hj_hk_1O~h3_{!. ~3!} For scalar QED described by the Lagrangian:

L521 4FmnF

mn_1~D

mf!†~Dmf!2l~f†f2m2!2, ~4!

with D_m5]_m1ieA_m andf5(f11if2)/

A

2, the connection of the configuration space is given by@7,12#

Gj k i ₅

H

i

j k

J

1Tj k

i

, ~5!

where$_{j k}i%is the Christoffel symbol of the field configuration space and Ti_{j k} is a quantity induced by generators of the gauge transformation. The Christoffel symbol $_jki% can be computed from the following metric tensor of scalar QED:

G_f

a(x)fb(y )5dabd

4_~x2y!, GA_m(x)A_n(y )52gmnd4~x2y!,

GA_m(x)f_a(y )50. ~6! According to Vilkovisky’s prescription@7#, the metric tensor of the field configuration space is obtained by differentiating twice with respect to the fields in the kinetic Lagrangian. For the above metric tensor, we have$j k

i_{%50 since each}

compo-nent of the tensor is field independent. However, the Christ-offel symbol would be non-vanishing if one parametrizes Eq. ~4! with polar variables r andx such that f15rcosx and

f25rsinx. Finally, to determine Tj k

i _{, let us specify}

genera-tors g_ai of the scalar-QED gauge transformations: g_yfa(x)52eab_ef

b~x!d4~x2y!,

g_yAm(x)52]_md4~x2y!, ~7! where eab is a skew-symmetric tensor with e1251. The quantity Ti_{j k}is related to the generators g_ai via@7#

T_{j k}i 52B_jaD_kg_ai11 2ga l_D lgb i_B j a_B k b_{1 j↔k, ~8!}

where B_ka5Nabgkb with Nab being the inverse of Nab [g_ak_g

b

l_G

kl. The expression for Tj k

i _{can be easily understood}

by realizing that i, j, . . . ,l are function-space indices, while a andb are space-time indices. Hence, for example,

D_f 1(z)gy A_m(x) 5]gy A_m(x) ]f1~z!1

H

A_m~x! jf1~z!

J

gy j , ~9!

where the summation over j also implies an integration over the space-time variable in the function j.

The one-loop effective action of scalar QED can be cal-culated from Eq. ~2! with each quantum fluctuation hi re-placed bysi_{(F,f). The result is written as@12#:}

G@F#5S@F#2i₂\lndetG1i\

2 lndetD˜i j

21_{, ~10!}

where S@F# is the classical action with F denoting generi-cally the background fields; lndetG arises from the function-space measure @Df#[)_xdf(x)

A

detG; and D˜_{i j}21 is the modified inverse propagator:

D˜_{i j}215 d 2_S

dFi_dFj2Gi j k_@F# dS

dFk. ~11!

To study the symmetry-breaking behavior of the theory, we focus on the effective potential which is obtained fromG@F# by setting each background fieldFi to a constant.

The Vilkovisky-DeWitt effective potential of scalar QED has been calculated in various gauges and different scalar-field parametrizations @11,12,14#. The results all agree with one another. In this work, we calculate the effective potential and other relevant quantities in the Landau-DeWitt gauge @15#, which is characterized by the gauge-fixing term: Lg f 52(1/2j)(]_mBm2ieh†F1ieF†h)2, with j→0. In Lg f,

Bm[Am2A_clm, and h[f2F are quantum fluctuations while A_clm andF are background fields. For the scalar fields, we further write F5(rcl1ixcl)/

A

2 and h5(r1ix)/

A

2.

The advantage of performing calculations in the Landau-DeWitt gauge is that T_jki vanishes@11# in this case. In other words, the Vilkovisky-DeWitt formalism coincides with the conventional formalism in the Landau-DeWitt gauge.

For computing the effective potential, we choose A_clm 5xcl50, i.e., F5rcl/

A

2. In this set of background fields,

Lg f becomes Lg f52 1 2j~]mB m_] nBn22erclx]mBm1e2rcl 2_x2_{!. ~12!} We note that B_m2x mixing in Lg f is j dependent, and

therefore would not cancel out the corresponding mixing term in the classical Lagrangian of Eq. ~4!. This induces mixed-propagators such as

^

0uT„A_m(x)x(y )_…u0

&

or

^

0uT„x(x)A_m(y )_…u0

&

. The Faddeev-Popov ghost Lagrangian in this gauge reads

LF P5v*~2]22e2rcl 2_!

v. ~13!

With each part of the Lagrangian determined, we are ready to compute the effective potential. Since we choose a field-independent flat-metric, the one-loop effective potential is completely determined by the modified inverse-propagators D˜_{i j}21 @16#. From Eqs. ~4!, ~11!, ~12! and ~13!, we arrive at

D˜_B mB_n 21 _5~2k2_1e2_r 0 2_!gmn₁

S

₁₂1 j

D

kmkn, D ˜ B_mx 21_5ikm_er 0

S

12 1 j

D

, D ˜ xx 21₅

S

_k2_2m G 2₂1 je2r0 2

D

_, D˜_rr215~k22m_H2!, D˜_v*v5~k22e2r0 2_!22 , ~14!

where we have set rcl5r0, which is a space-time independent constant, and defined m_G25l(r₀222m2), m_H25l(3r₀222m2). Using the definition G@r0# 52(2p)4d4(0)Ve f f(r0) along with Eqs.~10! and ~14!, and taking the limit j→0, we obtain Ve f f(r0)5Vtree(r0)

1V1-loo p(r0) with V1-loo p~r0!5 2i\ 2

E

dn_k ~2p!nln@~k22e2r0 2_!n23 3~k2_2m H 2_!~k2_2m 1 2_!~k2_2m 2 2_{!#, ~15!} where m₁2 and m₂2 are solutions of the quadratic equation (k2)22(2e2r₀21m_G2)k21e4r₀450. One notices that the gauge-boson’s degree of freedom in V1-loo phas been contin-ued to n23 in order to preserve the relevant Ward identities. For example, this continuation is crucial to ensure the Ward identity which relates the scalar self-energy to the con-tribution of the tadpole diagram. Our expression for

V1-loo p(r0) agrees with previous results obtained in the

unitary gauge @14#. One could also calculate the effect-ive potential in the ghost-free Lorentz gauge with L_{g f}5 2(1/2j)(]_mBm)2. The cancellation of the gauge-parameter (j) dependence in the effective potential has been demon-strated in the case of massless scalar QED where m250 @11,12#. It can be easily extended to the massive case, and the resultant effective potential coincides with Eq. ~15!. In the Appendix, we will also demonstrate the cancellation of gauge-parameter dependence in the calculation of Higgs-boson self-energy. The obtained self-energy will be shown to coincide with its counterpart obtained from the Landau-DeWitt gauge. We do this not only to show that the Vilkovisky-DeWitt formulation coincides with the ordinary formulation in the Landau-DeWitt gauge, but also to illus-trate how it gives rise to identical effective action in spite of beginning with different gauges.

It is instructive to rewrite Eq.~15! as

V1-loo p@r0#5 \ 2

E

dn21kW ~2p!n21„~n23!vB~kW!1vH~kW! 1v1~kW!1v2~kW!…, ~16! where vB(kW)5

A

kW21e2r0 2 , vH(kW)5

A

kW21mH 2 and v₆(kW) 5

A

kW21m₆2. One can see that V1-loopis a sum of the zero-point energies of four excitations with masses mB[er0, m_H, m₁ and m₂. Since there are precisely four physical degrees of freedom in the scalar QED, we see that the Vilkovisky-DeWitt effective potential does exhibit a correct number of physical degrees of freedom. Such a nice property is not shared by the ordinary effective potential calculated in the ghost free Lorentz gauge just mentioned. As will be shown later, the ordinary effective potential in this gauge contains unphysical degrees of freedom.

III. VILKOVISKY-DEWITT EFFECTIVE POTENTIAL OF THE SIMPLIFIED STANDARD MODEL

In this section, we compute the effective potential of the simplified standard model where charged boson fields and all fermion fields except the top-quark field are disregarded. The gauge interactions for the top quark and the neutral scalar bosons are prescribed by the following covariant derivatives @9#: D_mtL5

S

]m1igLZm2 2 3ieAm

D

tL, D_mt_R5

S

]_m1ig_RZ_m22 3ieAm

D

tR, D_mf5„]_m1i~g_L2g_R!Z_m…f, ~17! where Z_mand A_mdenote the Z boson and the photon respec-tively; the coupling constants gL and gR are given by gL 5(2g1/21g2/3) and gR5g2/3 with g15g/cosuW and g2 52etanuWrespectively. The self-interactions of scalar fields

are described by the same potential term as that in Eq. ~4!. Clearly this toy model exhibits a U(1)_A3U(1)_Z symmetry where each U(1) symmetry is associated with a neutral gauge boson. The U(1)_Zcharges of t_L, t_R andfare related in such a way that the following Yukawa interactions are invariant under U(1)A3U(1)Z:

LY52y t¯LftR2y t¯Rf*tL. ~18!

Since Vilkosvisky-DeWitt effective action coincides with the ordinary effective action in the Landau-DeWitt gauge, we thus calculate the effective potential in this gauge, which is defined by the following gauge-fixing terms @17#:

Lg f52 1 2a

S

]m˜Z m₁ig1 2 h †_F2ig1 2 F †_h

D

2 2₂1_b~]mA˜m!2, ~19! with a,b→0. We note that A˜m, Z˜m and h are quantum fluctuations associated with the photon, the Z boson and the scalar boson respectively, i.e., Am5A_clm1A˜m, Zm5Z_clm1Z˜m, and f5F1h with A_clm, Z_clm and F being the background fields. For computing the effective potential, we takeF as a space-time-independent constant denoted as ro, and set Aclm

5Zclm50. Following the method outlined in the previous

sec-tion, we obtain the one-loop effective potential

V_VD~r₀!5\ 2

E

dn21kW ~2p!n21„~n23!vZ~kW! 1vH~kW!1v1~kW!1v2~kW!24vF~kW!…, ~20! where v_i(kW)5

A

kW2_1m i 2 _{with m} Z 2_5(g 1 2_/4)r 0 2_{, m} 6 2_5m Z 2 11 2(mG 2_6m G

A

mG 2_14m Z 2

) and m_F2[m_t25y2r₀2/2. The Gold-stone boson mass mG is defined as before, i.e., mG

2_5l(r

o 2 22m2_{) withmbeing the mass parameter of the Lagrangian.} One may notice the absence of photon contributions in the above effective potential. This is not surprising since photons do not couple directly to the Higgs boson.

Performing the integration in Eq.~20! and subtracting the infinities with modified minimal subtraction ~MS! scheme prescription, we obtain VVD~r0!5 \ 64p2

S

mH 4 lnmH 2 k21mZ 4 lnmZ 2 k21m₁ 4 lnm1 2 k2 1m₂4_lnm2 2 k2 24mt 4_lnmt 2 k2

D

2 \ 128p2 3~3mH 4_15m Z 4_13m G 4_112m G 2 m_Z2212m_t4!, ~21! where k is the mass scale introduced in the dimensional regularization. Although V_VD(r₀) is obtained in the Landau-DeWitt gauge, we should stress that any other gauge with non-vanishing Ti_jk should lead to the same result. For later comparisons, let us write down the ordinary one-loop effec-tive potential in the Lorentz gauge@removing the scalar part of Eq.~19!# as follows @2#: VL~r0!5 \ 2

E

dn21kW ~2p!n21„~n21!vZ~kW!1vH~kW! 1va~kW,a!1vb~kW,a!24vF~kW!…, ~22!

where a is the gauge-fixing parameter and v_a,b(kW,a) 5

A

kW21m_a,b2 (a) with m_a2(a)51/2•(m_G2 1

A

m_G424am_Z2m2_G) and m_b2(a)51/2•(m_G2 2

A

m_G424am_Z2m_G2). It is easily seen that there are 6 bosonic degrees of freedom in V_L, i.e., two extra degrees of freedom emerge as a result of choosing the Lorentz gauge. In the Landau gauge, which is a special case of the Lorentz gauge witha50, there is still one extra degree of freedom in the effective potential @6#. Since the Landau gauge is adopted most frequently for computing the ordinary effective poten-tial, we shall takea50 in VL hereafter. Performing the

in-tegrations in VL and subtracting the infinities, we obtain

VL~r0!5 \ 64p2

S

mH 4 lnmH 2 k213mZ 4 lnmZ 2 k2 1mG 4_lnmG 2 k224mt 4_lnmt 2 k2

D

2₁₂₈\_p2~3mH 4_15m Z 4_13m G 4_212m t 4_{!. ~23!} We remark that VL differs from VVD except at the point of

extremum wherer₀252m2. At this point, one has m_G250 and m₆25m_Z2, which lead to VVD(r052m2)5VL(r0

2_52m2_).

That VVD5VL at the point of extremum is a consequence of

the Nielsen identity@3# mentioned earlier.

To derive the Higgs boson mass bound from VVD(r0) or VL(r0), one encounters a breakdown of the perturbation theory at the point of sufficiently large r0 such that, for instance, (l/16p2)ln(lr₀2/k2).1. To extend the validity of the effective potential to the large-r0 region, the effective potential has to be improved by the renomalization group ~RG! analysis. Let us denote the effective potential as Ve f f

which includes the tree-level contribution and quantum cor-rections. The renormalization-scale independence of Ve f f

im-plies the following equation @18,5#:

S

2m~gm11!_]m] 1bgˆ ] ]gˆ2~gr11!t ] ]t14

D

3Ve f f~tr0 i ,m,gˆ,k!50, ~24!

wherem is the mass parameter of the Lagrangian as shown in Eq.~4!, and bgˆ5k dgˆ dk, gr52k dlnr dk , gm52k dlnm dk , ~25!

with gˆ denoting collectively the coupling constantsl, g1,g2 and y;r₀i is an arbitrarily chosen initial value forr0. Solving this differential equation gives

Ve f f~tr0 i ,mi,gˆi,k!5exp

S

E

0 lnt ₄ 11g_r~x!dx

D

3Ve f f„r0 i_,m~t,m i!,gˆ~t,gˆi!,k…, ~26! with x5ln(r0

8

/r0 i

) for an intermediate scaler0

8

, and tdgˆ

dt5

bgˆ„gˆ~t!…

m~t,mi!5miexp

S

2

E

0

lnt₁1g_m~x!

11g_r~x!dx

D

. ~28! To fully determine Ve f f at a larger0, we need to calculate the b functions of l, g1, g2 and y, and the anomalous di-mensions g_m and g_r. It has been demonstrated that the n-loop effective potential is improved by the (n11)-loopb andg functions @19,20#. Since the effectve potential is cal-culated to the one-loop order, a consistent RG analysis re-quires the knowledge ofb andg functions up to a two-loop accuracy. However, as the computations of two-loopb and g functions are quite involved, we will only improve the tree-level effective potential with one-loop b and g func-tions. After all, the main focus of this paper is to show how to obtain a gauge-independent Higgs boson mass bound rather than a detailed calculation of this quantity.

To compute one-loopbandgfunctions, we first calculate the renormalization constants Z_l,Zg₁, Zg₂,Zy,Zm2 and Z_r,

which are defined by lbare_5Z ll, g1 bare_5Z g₁g1, g2 bare_5Z g₂g2, ybare5Zyy , ~m2!bare5Zm2m2, rbare5

A

Z_rr.

~29! In the ordinary formalism of the effective action, all of the above renormalization constants except Z_r are in fact gauge-independent at the one-loop order in the MS scheme. For Z_r, the result given by the commonly adopted Landau gauge differs from that obtained from the Landau-DeWitt gauge. In Appendix A, we shall reproduce Z_robtained in the Landau-DeWitt gauge with the general Vilkovisky-DeWitt formulation. The calculation of various renormalization con-stants are straightforward. In the MS scheme, we have ~we will set\51 from this point on!

Z_l512 1 128p2e

8

S

3g₁4 l 224g1 2₂16y 4 l 132y21160l

D

, Zg₁5Zg₂512 1 216p2e

8

S

27g₁2 8 12g2 2_23g 1g2

D

, Zy511 1 192p2e

8

~9g1 2_14g 1g2224y2!, Z_m2511 1 128p2e

8

S

3g₁4 l 212g1 2₂16y 4 l 116y2196l

D

, Z_r511 1 32p2_e

₈

~25g1 2_14y2_{!, ~30!} where 1/e

8

[1/e112 gE2 1

2 ln(4p) withe5n24. The

one-loopb and g functions resulting from the above renormal-ization constants are

bl5 1 16p2

S

3 8g1 4_23lg 1 2_22y4_14ly2_120l2

D

_, bg₁5 g1 4p2

S

g₁2 162 g1g2 18 1 g₂2 27

D

, bg₂5 g2 4p2

S

g₁2 162 g1g2 18 1 g₂2 27

D

, by5 y 8p2

S

y 2₂3g1 2 8 1 g1g2 12

D

, gm5 1 2p2

S

3l 4 1 3g₁4 128l 2 3g₁2 32 2 y4 8l 1 y2 8

D

, gr5 1 64p2~25g1 2_14y2_{!. ~31!}

Similar to what was mentioned earlier, all of the above quan-tities are gauge-independent in the MS scheme except g_r, the anomalous dimension of the scalar field. In the Landau gauge of the ordinary formulation, we have

gr5

1

64p2~23g1

2_14y2_{!. ~32!}

IV. THE HIGGS BOSON MASS BOUND

The lower bound of the Higgs boson mass can be derived from the vacuum instability condition of the electroweak ef-fective potential@21#. In this derivation, there exists different criteria for determining the instability scale of the elec-troweak vacuum. The first criterion is to identify the insta-bility scale as the critical value of the Higgs-field strength beyond which the renormalization-group- ~RG-! improved tree-level effective potential becomes negative @22–24#. To implement this criterion, the tree-level effective potential is improved by the leading @24# or next-to-leading order @22,23# renormalization group equations, where one-loop or two-loopb andgfunctions are employed. Furthermore, one-loop corrections to parameters of the effective potential are also taken into account @23,24#. However, the effect of one-loop effective potential is not considered.

To improve the above treatment, Casas et al.@25# consid-ered the effect of RG-improved one-loop effective potential. The vacuum-instability scale is then identified as the value of the Higgs-field strength at which the sum of tree-level and one-loop effective potentials vanishes. In our subsequent analysis, we will follow this criterion except that the one-loop effective potential is not RG improved.

To derive the Higgs boson mass bound, one begins with Eq. ~26! which implies

V_tree~tr₀i,m_i,l_i!51

4x~t!l~t,li!„~r0

i_!2_22m2_~t,m i!…2,

with x(t)5exp„*₀lnt@4/11g_r(x)#dx…. Since Eq. ~28! implies that m(t,m_i) decreases as t increases, we then have V_tree(tr₀i,m_i,l_i)'1

4x(t)l(t,li)(r0

i₎4 _for a sufficiently large t. Similarly, the one-loop effective potential V1-loop(tr0

i

,mi,gˆi,k) is also proportional to

V1-loop„r0 i_,m(t,m

i),gˆ(t,gˆi),k… with the same proportional

constantx(t). Because we shall ignore all running effects in

V1-loop, we can take gˆ(t,gˆi)5gˆi and m(t,mi)5(1/t)mi in

V_1-loop. For a sufficiently large t, V1-loo p can also be

ap-proximated by its quartic terms. In the Landau-DeWitt gauge with the choicek5r₀i, we obtain

V_VD'~r0 i_!4 64p2

F

9li 2_ln~3l i!1 g_1i4 16ln

S

g_1i2 4

D

2yi 4 ln

S

yi 2 2

D

1A1 2 ~g1i,li!lnA1~g1i,li! 1A₂2_~g 1i,li!lnA2~g1i,li! 23₂

S

10l_i21lig1i 2₁ 5 48g1i 4_2y i 4

D

G

, ~34! where A₆(g₁,l)5g₁2/41l/2•(16

A

11g₁2/l). Similarly, the effective potential in the Landau gauge is given by

VL' ~r0 i_!4 64p2

F

9li 2 ln~3li!1 3g_1i4 16 ln

S

g_1i2 4

D

2yi 4 ln

S

yi 2 2

D

1li 2 ln~li!2 3 2

S

10li 2_1l ig1i 2₁ 5 48g1i 4_2y i 4

D

G

. ~35! Combining the tree-level and the one-loop effective poten-tials, we arrive at Ve f f~tr0 i ,mi,gˆi,k!' 1 4x~t!„l~t,li!1Dl~gˆi!…~r0 i_!4_, ~36! whereDl represents the one-loop corrections obtained from Eqs. ~34! or ~35!. Let tVI5rVI/r0

i

, the condition for the vacuum instability of the effective potential is then @25#

l~tVI,li!1Dl~gˆi!50. ~37!

We note that the couplings gˆi in Dl are evaluated at k 5r0

i

, which can be taken as the electroweak scale. Hence we have g1i[g/cosuW50.67, g2i[2etanuW50.31, and yi 51. The running coupling l(tVI,li) also depends upon

g1,g2and y throughbl, andgr shown in Eq.~31!. To solve Eq. ~37!, we first determine the running behaviors of the coupling constants g₁,g₂ and y. For g₁ and g₂, we have

td„gl 2 ~t!… dt 52gl~t! bg_l„gˆ~t!… 11g_{r„gˆ~t!…}'bgl 2_, ~38!

where l51,2, and the contribution ofg_r is neglected in ac-cordance with our leading-logarithmic approximation. Also bg

l

25g_l2/2p2•(g₁2/162g₁g₂/181g₂2/27). Although the

dif-ferential equations for g₁2 and g₂2 are coupled, they can be easily disentangled by observing that g₁2/g₂2 is a RG-invariant. Numerically, we have b_g

l

25c_lg_l4 with c₁52.3

31023 _{and c}

251.131022. Solving the differential equa-tions gives

g_l22~t!5g_l22~0!2cllnt. ~39!

With g₁(t) and g₂(t) determined, the running behavior of y can be calculated analytically @4#. Given by2[2yb_y5c₃y4

2c4g1 2

y2with c352.531022and c458.531023, we obtain

y2~t!5

FS

g1 2 ~t! g_1i2

D

c₄/c₁

S

y_i222 c3 c₁1c₄g1i22

D

1 c3 c11c4 g₁22~t!

G

21 . ~40!

Now the strategy for solving Eq.~37! is to make an initial guess onli, which enters intol(t) and Dl, and repeatedly

adjustliuntill(t) completely cancels Dl. For tVI5102~or

r0'104 GeV) which is the new-physics scale reachable by the CERN Large Hadron Collider ~LHC!, we find li54.83 31022 _{for the Landau-DeWitt gauge, and l}

i54.831022

for the Landau gauge. For a higher instability scale such as the scale of grand unification, we have tVI51013 or r0 '1015 _{GeV. In this case, we find l}

i53.1331021 for both

the Landau-DeWitt and Landau gauges. The numerical simi-larity between the li of each gauge can be attributed to an

identical b function for the running of l(t), and a small difference between the Dl of each gauge. We recall from Eq.~27! that the evolutions of l in the above two gauges will be different if the effects of next-to-leading logarithms are taken into account. In that case, the difference between the grof each gauge gives rise to different evolutions forl. For

a large tVI, one may expect to see a non-negligible

differ-ence between thel_iof each gauge.

The critical valueli54.8331022corresponds to a lower

bound for the MS mass of the Higgs boson. Since mH 52

A

lm, we have (mH)MS>77 GeV. For li53.1331021,

we have (mH)MS>196 GeV. To obtain the lower bound for the physical mass of the Higgs boson, finite radiative correc-tions must be added to the above bounds @4#. We will not pursue these finite corrections any further since we are sim-ply dealing with a toy model. However we would like to point out that such corrections are gauge-independent as en-sured by the Nielsen identity @3#.

V. CONCLUSION

We have computed the one-loop effective potential of an Abelian U(1)3U(1) model in the Landau-DeWitt gauge, which reproduces the result given by the gauge-independent Vilkovisky-DeWitt formulation. One-loopb andg functions

were also computed to facilitate the RG improvement of the effective potential. A gauge-independent lower bound for the Higgs-boson self-coupling or equivalently the MS mass of the Higgs boson was derived. We compared this bound to that obtained using the ordinary Landau-gauge effective po-tential. The numerical values of both bounds are almost iden-tical due to the leading-logarithmic approximation we have taken. A complete next-to-leading-order analysis should bet-ter distinguish the two bounds. This improvement as well as extending the current analysis to the full standard model will be reported in future publications.

Finally we would like to comment on the issue of com-paring our result with that of Ref. @5#. So far, we have not found a practical way of relating the effective potentials cal-culated in both approaches. In Ref.@5#, to achieve a gauge-invariant formulation, the theory is written in terms of a new set of fields which are related to the original fields through non-local transformations. Taking scalar QED as an ex-ample, the new scalar field f

8

(xW) is related to the original field through@6#

f

8

~xW!5f~xW!exp

S

ie

E

d3y AW~yW!•¹W_yG~yW2xW!

D

, ~41! with G( yW2xW) satisfying ¹2G(yW2xW)5d3(yW2xW). To our knowledge, it does not appear obvious how one might incor-porate the above non-local and non-Lorentz-covariant trans-formation into the Vilkovisky-DeWitt formulation. This is an issue deserving further investigations.

ACKNOWLEDGMENTS

We thank W.-F. Kao for discussions. This work is sup-ported in part by National Science Council of R.O.C. under grant numbers NSC 87-2112-M-009-038, and NSC 88-2112-M-009-002.

APPENDIX: THE HIGGS-BOSON SELF-ENERGY AND VILKOVISKY-DEWITT EFFECTIVE ACTION

In this Appendix, we calculate the Higgs-boson self-energy of scalar QED from the Vilkovisky-DeWitt effective action. We will focus on the momentum-dependent part of the self-energy, which is not a part of the effective potential calculated in Sec. II. Furthermore only the infinite part of the self-energy will be calculated. We thus perform the calcula-tion in the symmetry phase of the theory.

We begin with the Lagrangian in Eq. ~4! where m2 is negative, i.e.,2m2_[u2_{.0. In this case,f}

1andf2 have an identical mass m_f252lu2. Let us renamef1asr andf2as

x according to our notation in the symmetry-broken phase. If one follows the background field expansion in Eq. ~2!, one would expand the QED action by writing Am5A_clm1Bm, r 5rcl1h1, andx5xcl1h2, with Aclm, rcl andxcl the

clas-sical background fields, and Bm, h1 andh2 the correspond-ing quantum fluctuations. However, as mentioned earlier, the above quantum fluctuations should be replaced by vectorssi

in the configuration space. Hence the action in Eq.~2! should be expanded covariantly@7# in powers ofsi. To simplify our

notations, we use B˜_m and h˜i to denote the new quantum

fluctuations. Since we will only calculate the self-energy of r, we may take A_clm5xcl50 for simplicity. With the

covari-ant expansion, the Lagrangian in Eq. ~4! generates the fol-lowing quadratic terms:

Lquad52 1 4~]mB˜n2]nB˜m! 2₁1 2~]mh˜1!~] m_h˜ 1! 11 2~]mh˜2!~] m_h˜_2!1ercl~]m_h˜_2!B˜m2eh˜_2~]m_rcl!B˜m 11 2e 2_r cl 2_B˜m_B˜ m2l

F

1 2rcl 2_~3h˜ 1 2_1h˜ 2 2_!1u2_~h_˜ 1 2_1h˜ 2 2_!

G

2G_FFm_Fn l dS dFlf˜mf˜n, ~A1!

whereFlandf˜ldenote generically the classical background fields and the quantum fluctuations respectively. We choose the R_j background-field gauge with the gauge-fixing term:

L_{g f}52 1 2a~]mB˜

m_2aer

clh˜2!2. ~A2! The corresponding Faddeev-Popov Lagrangian is then

L_{F P}5v*~2]22ae2r_cl2!v. ~A3! Compared to the usual background-field formalism, the qua-dratic quantum flucuations Lquadcontain extra terms

propor-tional to the connectionGi_{j k}of the configuration space. These extra terms are crucial for the cancellation of gauge-parameter dependence in the Higgs-boson self-energy. From Eqs. ~5!, ~6!, ~7! and ~8!, we calculate those connections which are relevant to the Higgs-boson self-energy. We find

GAr(z)_m(x)A_n( y )uF52e2rcl~z!~]xmNxz!~]ynNy z! G_x(x)x(y)r(z) uF5e2Nxy„d4~y2z!1d4~x2z!…rcl~z!

2e4_r

cl~z!NzxNzyrcl~x!rcl~y! GAr(z)_m(x)x(y)uF5e~]mxNzx!d4~z2y!2e3rcl~z!

3~]xmN xz_!Nz y_r cl~y!, ~A4! where Nx y5

K

x

U

1 ]2_1e2_r cl 2 ~X!

U

y

L

with X_mux

&

5x_mux

&

; and the notationu_F denotes evaluating the connection at the classical background fields. The above connections are to be multiplied by dS/dru_F[(2]2 22lu2_2lr

cl 2

)rcl with the space-time variable z

inte-grated over. It is interesting to note that the product ofG and dS/dr contain terms which are able to generate the Higgs-boson self-energy. For example, in the expression 2*d4_xd4_{y (G}

A_m(x)A_n( y )

r(z) _dS/dru

rcl50 in Nxzand Ny z and contract the pair of gauge fields.

This gives rise to, in the momentum space, the following Higgs-boson self-energy: S_rAA_~p2_!52ae 2 8p2 1 e

8

p 2_{, ~A5!} where 1/e

8

[1/e11 2 gE2 12ln(4p) with e5n24. For 2*d4_xd4_{y (G} x(x)x(y) r(z) _dS/dru F)h˜2(x)h˜2(y ), we again set rcl 50 in Nx y _{and contract the pair of scalar fields. We obtain}

the Higgs-boson self-energy S_rxx_~p2_!5 e 2 4p2 1 e

8

p 2_{. ~A6!}

Finally, the term 2*d4xd4y (GAr(z)_m(x)x(y)dS/

druF)B˜m(x)h˜2(y ) can produce an effective rcl2B˜m2h˜2 vertex, namely, @*d4pd4k/(2p)8#Gm( p,k)r_cl(k)B˜_m( p)h₂ (2p2k), with G_m( p,k)5i(pm/ p2)_•(k222lu2). This ver-tex can contribute to the Higgs-boson self-energy by con-tracting with another vertex of the same kind. Similarly, it could contract with an ordinaryrcl2B˜m2h˜2 vertex. It turns out that both contractions produce only finite contributions to the momentum-dependent part of the Higgs-boson self-energy. Therefore Eqs.~A5! and ~A6! are the only divergent contributions ofG_{j k}i to the momentum-dependent part of the

Higgs-boson self-energy. These contributions are to be added to the self-energy obtained by contracting a pair of ordinaryrcl2B˜m2h˜2 vertices. We find

S_rordinary_~p2_!5 e 2 8p2 1 e

8

~31a!p 2_{. ~A7!}

From Eqs.~A5!, ~A6! and ~A7!, we arrive at Sr~p2!5SrAA~p2!1Srxx~p2!1Srordinary~p2!5 5e2 8p2 1 e

8

p 2_. ~A8! We can see that the gauge-parameter dependence of S_rordinary

( p2) is cancelled by that ofS_rAA( p2). From Eq.~A8!, the wave-function renormalization constant of the Higgs bo-son is found to be Z_r5125e 2 8p2 1 e

8

. ~A9!

This result can be applied to the model in Sec. III with the replacement e→2g1/2 according to Eq.~17!. Hence Zr51 2(5g1

2

/32p2)(1/e

8

) in that model, which reproduces the rel-evant part of Eq. ~30! calculated in the Landau-DeWitt gauge.

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(F,f), as suggested in Ref. @7#, is not satisfactory for cal-culations beyond the one-loop order. An effective action ob-tained by such a prescription does not generate one-particle irreducible diagrams at the higher loops. A modified

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