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The lower and upper bounds on the mass of the Higgs boson in SU(2) gauge theory
Volume 200, number 1,2
PHYSICS LETTEI~SB
7 January 1988
THE LOWER AND UPPER BOUNDS ON THE MASS OF THE HIGGS BOSON IN SU(2) GAUGE THEORY Guang-Jiong NI
J
International Centre for Theoretical Physics, 1-34100 Trieste, Italy
Sen-Yue L O U and Su-Qing C H E N Physics Department, Fudan University, Shanghai, P.R. China
Received 14 August 1987
Using the method of the gaussian effective potential, we analyze the Higgs mechanism of SU(2) gauge theory in the quantum version. We obtain an estimation of lower and upper bounds on the mass of the Higgs scalar boson 0.925mw< mn < 2.07row. If the mass of the charged vector boson row= 82 GeV, then 75.9 GeV < mH< 170 GeV.
The standard model o f electroweak and strong interactions is healthier than it ever was [ 1 ]. However, m a n y problems still remain open. Among them the origin o f the Higgs mechanism and the mysterious Higgs particle attract much attention, ls the Higgs particle really an elementary particle or rather a composite one? Where are the Higgs and top quarks? It is speculated [2] that the most important goal o f experimental particle physics in the next decade is the clarification o f the electroweak symmetry breaking. The solution must be within the TeV energy region. If an experiment can detect a Higgs of 1-2 TeV, then either (a) the Higgs is found; and/or (b) new physics is found; or (c) at least, the onset of the new regime of weak interactions becoming strong can be studied. More quantitatively, the Higgs mass mH is classified into four regions [3]: (i) 10 G e V < m H < 150 GeV, no problems for the standard model up to energies > M p ( ~ 1019 GeV); (ii) 150 G e V < m H < 6 0 0 GeV, information about new physics at a scale below Mp is possible; (iii) mH > 600 GeV, new physics around or below 1 TeV, the standard model description breaks down; (iv) mH < 10 GeV, interesting information about dilatation symmetry is implied. Experimentally, there are almost no reliable lower limits on the Higgs mass [2], while theoretically, there exist some estimations of the lower and upper bounds on mH. The lower bound is constrained by the v a c u u m stability which is usually analyzed in terms of an effective potential calculated to the one-loop level, including both the contributions o f bosons and fermions. On the other hand, the upper bound on mH, mH, is based on requiring the perturbation regime to be valid up to some large energy scale A [ 2 - 9 ]. The value of tnH decreases with the increase of A. Being a physical cut-off, A also appears in the so-called "triviality" problem of the real j.~4 model [ 10-12 ]. It was pointed out that for a stability analysis the A dependence is inevitable whereas the perturbative theory may lose its way. Thus, a variational method in q u a n t u m field theory, the gaussian effective potential ( G E P ) approach, attracts much attention in recent years [ 10-14]. In a previous paper [ 14], we discussed in detail the triviality problem of the ,~4 model. It is shown that after quantization by path integration, the potential in the lagrangian, J On leave of absence from Physics Department, Fudan University, Shanghai, P.R. China. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
161
Volume 200, number 1,2
PHYSICS LETTERS B
V(O) = - ½ao~}2 + (1/4!)2o~} 4 ,
7 January 1988 (1)
evolves into an effective potential U ( ~ ) which has a stable minimum, i.e., a broken phase so long as the following criteria are satisfied: 0 < 2 o <4/J2(#/A) ,
ao > qcrAz ,
(2,3)
where
J2(/2/A) = (1/2re 2) {ln[A//x+ ,/(A/#) 2 + 1] - ( A / / 2 ) / ~
+ 1},
~/c~= ( Z c / ~ c 2 + 1 ) 2o/16n 2 <2o/16re 2 ,
(4) (5)
with Zc = ½exp(8n2/2o + 1 ) >> 1 ,
(6)
and A being a cut-off o f the three-dimensional m o m e n t u m . In this letter the GEP method will be used to discuss the S U ( 2 ) gauge theory coupled to an isospinor of complex scalar fields. The Higgs mechanism in this model enables us to obtain the lower and upper bounds for the mass of Higgs particle to be 0.925mw < mH < 2.07row,
(7)
mw being the mass of the charged vector boson. Beginning from the lagrangian density
£P(x) = - ¼G~a"G,,~,~+ (D~,¢) + (D~'q$) + ao~ + q$- (1/3!)2o(~ + ¢~)2,
(8)
with P v ,u /t v G~/ t / ) =0 ¢t Aa-O A~-g%bcAbAc
,
DF,~=[O~+iA~,~(x)½r~]O(x) ,
(9,10)
and \02/'
0+ = (¢}*0") '
(ll)
we redefine
W~=(I/x/2)(Ag +iAS) , Z~=A~
(12,13)
to represent the charged and neutral massive bosons, respectively. But later on for convenience in calculation, sometimes we shall still use the real A ~ and A2. Also, through a gauge transformation, one can always simplify the isospinor (11 ) as = ~
'(7)
,
(14)
where ~ being a real scalar field with the other three degrees of freedom in (11 ) being absorbed into the Iongitudinal polarizations o f massive W and Z bosons. We write down the hamiltonian of model 8) as usual: H = J - d3x[ ~(H~ 1 2 +H~, +HL
(15)
+H~) + Vl,
where
H¢=OL,e/O~=~, II~8=05HOh,=-h ', 162
(BZ=A~,A~,Z').
(16,17,18)
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
In (15) we have chosen the temporal gauge B ° = 0 so that H 2, =H;A,H;A. etc., and V= 1 (0i~)2 "t- 1 [( OiAij)2 'b ( OiA2j) 2 --O;A~ Oj A i l - O;A~OjA2;] + ½[ (0;Z:) 2 -OiZjOJZ;] -gZz(AzjA ~-Alj A~)
+ g2 Z;Z; W j* W j - g2 Z;Zj W i* W; + gZiJ(A2,A 1;-A2jA~ i) + lg 2(A ~AJi -AJ2A ~)(Az;A lj -AzjA,i) _ ½ao~2 + (1/4!)20~, - ¼g2 ~ Wi~2 _ ~gZ'~O;~- ~g2ZiZ,~2
(A'{ = O;A~ - OJA ~, Z ° = O;Zj - OJZ ', etc.) .
(19)
As in ref. [ 14 ], we quantize this model by evaluating the expectation value of H in a gaussian wave functional (ansatz):
~ = N e x p ( i f P¢,+i f pimlA] +i f P!42A~+i f P;zZ ' x
~:
X
x
_2
1
1f -'0 '
_1
f x y
f [A~(x)-Ai(x)]Fx,,(ff/)[A'l(y)-ff.~(y)]-~ ,f
xy
E=(TIHI~P)
[Ai(x)-Ai(x)lf~v(ff')[Ai(y)-21(y)l
ALl2
)
(20)
= f# x
= f ( ¼f,-.,-(~)+ ~b~,-x(2) + 31~.,.(W) --~If. Ox,,V2f~,l(~)_l - z f ~ x ,-. V ~ f --Tl(7~)_ f,~~. . .v.2.r._.l ( ~v,/ ~ .v
y
12
)
y
+ g2 Z Z ~ ff/j -g2 2,2j ~ ~ + g2[ 3Fz, I ( ITV) F z,J ( Z ) + Z;Z, F z~I ( ff/) + W~;W, F z~' ( 2 ) ] + lg2( ~ Wj - ~ W;)( ~ ~ - ~ IZV,)+g2Fz,) (liE) ~ UVi+ 3g2[Fz,~ ( I~)] 2 - ½ao [(2 + ½F~' (()] + (1/4!)2o {¢4 + 3Fz,.I ( ( ) ~ 2 + ] [Fz~' (()1 z }
_1_~g2[ ~/i[/~ ' -t- ~F ~,1( W)] [~2 -t- ½F2~' (~)1 + Ig2[ 2;2, + 3F.,7~' ( 2 ) 1 [ ¢ 2 + i F L ' ( ( ) ] )
J
(21 )
Then the variational procedures as in ref. [ 14] lead to F~12(l]) = C 3 jf ~ d3p x/p2-t-## 2 exp[iCBp-(x--y)] ,
(22)
d3p F~l(]~) = C 3 f (2z0 3~
(23)
1
exp[iCBp-(x--Y)l,
with ( B = W , Z, ~), Cw=Cz=x/3/-2=C ~/3, C¢= 1, fyF~12(B)FTfl (13)=~ .... The mass parameters .//4 n can be determined by minimizing the energy density # of the system
OglO~'~ I.,4 =;,-~= 0 , ]2~ = _ 0.o _t_~2o[~ I ~ + ~lI i ( / t ~ ) l + ½ g ~[3C11(]A2).. F I~,W,] +~g , 2 [sCII(I~)+ 3 Z;Z;]
(24)
/x2 = 2g 2 CI~ (It 2) + 4g2 I~ lfz; + ~g2[ ~ + ½I. (U ~)],
(25)
#2 =g2 CI, (¢t~) + ~gZ 2,Z, + ~g2( ~ W;) + lgZ [~: + ½11(/~)] +g2 CI. (#~v) •
(26) 163
Volume 200, number 1,2
PHYSICS LETTERSB
7 January 1988
Notice, however,/* 2 (B = W, Z, ~) is not the mass squared (m 2) of particle B, the latter should be the curvature of the effective potential at the stable broken vacuum. It is easy to prove that the broken vacuum is located at - / Z- t v a c~ - Wvac =0,
(27)
1 ~ac = (6/2o) [ ao -- Z).oI, (U~2 Ivac) -- ]g2CI~(/*2lvac)-]Cg2I,(/*~lva~)]=(3/2o)/*~lva~=_(3/)~o)/*~o,
(28)
while the mass squared of the observable particles are
m 2 w = f [ C I l ( m 2 w ) + C I ~ ( m ~ ) + ~ I ~ (/*¢o2)+ (3/42o)/*~o] ,
(29)
m 2 7 = f [ 2 C i l ( m ~ v ) + ~I~ ~ (/*¢o) 2 + (3/42o)/*~o],
(30)
and m2 =~v2ac().o + b \3 a
I2(/*~o)(J.oa+b) 2 "] a{a[8+2oI2(~o)]+bI2(/*~o)}]'
(31)
where
a = 2 + Cg212 ( m ~v) - Ce gH2 ( m 2 ) I2 ( m~ ) , 3 2 3 2 b= ~g C[ ag C I 2 ( m z2 ) I z ( m w2 ) - I 2 ( m w2 ) - 1/z(m2)] •
It is interesting to see that the mass squared of the W boson or Z boson coincides with/,2 Lva~ or f z I.... but this is not true for the Higgs particle. In all above expressions the integral I~ (/*2) reads: d3p 1 I, (/*~) = f (2g)3 X / ~ - 5 - ~ =U2JI (/*rflA)
=/*~(1/4~2){(A//*~)x/(A/my + 1 --ln [A//*. +,J(A//*~) 2 + 1]}.
(32)
We are not in a position to search for some relations among mw, mz and m . . First, the difference between (29) and (30) can be recast into the following form:
m 2 - m 2 = C f [I, (m~) - I , ( m 2 ) ] - (l/4n 2) Cg2[ m 2 ln(2A/mw) - m27 ln(2A/mz)] = ½Cg2[
2 2 mwI2( mw) - m z l22 ( mz) ] ,
(33)
where the approximation
ln( 2A/mw) =ln( 2A/mz) >> 1
(34)
has been made while eq. (32) and 12=.12 with (4) have beenused. Furthermore, the right-hand side of (33) can be written as
( l/4r~2)Cg2( m~v - m ~ ) ln( 2 A / m , ) .
(35)
If we assume m w = 8 2 GeV, m z = m w ( 1 + e ) , e>0, then m , = m w [ 1 +O(~)1 •
(36)
Hence, the same approximation as in (34) I2(m 2 ) -~I2(m~) = Ie(/*~o) = I 2 ( m ,2 ) ,
(37)
leads to
Cg212(m2,) = 2 . 164
(38)
Volume 200, number 1,2
PHYSICS LETTERSB
7 January 1988
Substitution of (37) with (38) into the expression of mH, eq. (31) yields
02 ~/0~ 2 Ivac= m ~ --~] f (~ac(Ao/f - 3.22)(0.642--2o/f)/(7.16+ 2 o / f ) ,
(39)
so the criterion for stability of the broken phase implies 0.642 <2o/g 2 < 3.22.
(40)
On the other hand, there is a well-known relation between mH and mw at the tree level [see eq. (19)] [8,9]
m21m2w = ~2olf .
(41)
A combination of (40) with (41) leads to 0.925 < mH/rnw < 2.07,
(42)
which accomplishes the proof of (7). If m w = 82 GeV, one obtains the lower and upper bounds for the mass of the Higgs particle: 75.9 G e V < mrj < 170 GeV.
(43)
It is interesting to see that our upper bound is not far from that of refs. [9,7], where f f t H = ( l / x f 2 ) ( 3 +
, ~ ) ~/2mw~ 135 GeV and r~H = 125 GeV are given, respectively. On the other hand, the lower bound in (43) seems much higher than that in ref. [4]. Although a series of difficulties has to be overcome before one can manipulate a realistic S U ( 2 ) × U ( I ) model and take the effect of fermions into account, our treatment does share the advantages of unity in point of view. We treat both the lower and upper bounds by the vacuum stability condition as shown in eq. (39). Besides (20), only a further approximation (33) or (34) is made. Eqs. ( 3 6 ) - ( 3 8 ) imply that the accuracy of the estimation would not be too bad so long as the cut-offA is large and the coupling constant g2 is small. Since the Yukawa coupling is even weaker than g2, we expect that the fermions will play a minor role in enhancing the instability of the vacuum till the mass of the fermion becomes very heavy. Correspondingly, we tentatively combine our lower-upper bounds with the plot in ref. [ 7] and are able to estimate the mass of the top quark: rnt> 125 GeV.
(44)
As recent experiments, especially the measurements on B°-13° mixing, imply that the top quark has a large mass [1,15,16], the estimated value in (44) may be also interesting. Hopefully, both (43) and (44) could serve as some test of the standard model. This work is partially supported by the NSF of China under Grant NO.KR12040. One of the authors (GuangJiong Ni) has benefitted from the lectures given at the Conference on Phenomenology in High Energy Physics (20-22 July 1987) and the Meeting on Search for Scalar Particles: Experimental and Theoretical Aspects (23, 24 July 1987) held at the International Centre for Theoretical Physics, Trieste, where this work was completed. He would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References [ l ] P. Darriulat, Lectureat Europian Intern. High energyphysicsConf. (Uppsala, Sweden, 25 June-1 July 1987). [2 ] G. Altarelli, Lectureat Meetingon Search for scalar particles: experimental and theoretical aspects (Trieste, Italy, 23, 24 July 1987). [3] C. Wetterich, Lecture at Meeting on Search for scalar particles: experimental and theoretical aspects (Trieste, Italy, 23, 24 July 1987). [4] N. Cabibbo, L. Maiani, G. Praisi and R. Petronzio, Nucl. Phys. B 158 (1979) 295. [5] L Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B 136 (1978) 115. 165
Volume 200, number 1,2
PHYSICS LETTERS B
[6] R. Dashen and H. Neuberger, Phys. Rev. Lett. 50 (1983) 1897. [7] M.A.B. Beg, C. Panagiotakopoulos and A. Sirlin, Phys. Rev. Left. 52 (1984) 883. [8] D.J.E. Callaway, Nucl. Phys. B 233 (1984) 189. [9] D.J.E. Callaway and R. Petronzio, Nucl. Phys. B 240 (1984) 577; B 267 (1986) 253; B 277 (1986) 50. [ 10] P.M. Stevenson, B. Alles and R. Tarrch, Phys. Rev. D 35 (1987) 2407, and references therein. [ 11 ] M. Consoli and G. Passarino, Phys. Lett. B 165 (1985) 113. [ 12] Y. Brihaye and M. Consoli, Nuovo Cimento 94A (1986) 1. [ 13] R. Ingermanson, Nucl. Phys. B 266 (1986) 620. [ 14] G,J. Ni, S.Y. Lou and S.Q. Chen, Fudan University preprint (1987), [ 15] A. Ali, Lecture at Conf. on Phenomenology in high energy physics (Trieste, Italy, 20-22 June 1987). [ 16] D,S. Du and Z.Y. Zhao, Princeton University preprint IASSNS-HEP-87/23 (April 1987).
166
7 January 1988
PHYSICS LETTEI~SB
7 January 1988
THE LOWER AND UPPER BOUNDS ON THE MASS OF THE HIGGS BOSON IN SU(2) GAUGE THEORY Guang-Jiong NI
J
International Centre for Theoretical Physics, 1-34100 Trieste, Italy
Sen-Yue L O U and Su-Qing C H E N Physics Department, Fudan University, Shanghai, P.R. China
Received 14 August 1987
Using the method of the gaussian effective potential, we analyze the Higgs mechanism of SU(2) gauge theory in the quantum version. We obtain an estimation of lower and upper bounds on the mass of the Higgs scalar boson 0.925mw< mn < 2.07row. If the mass of the charged vector boson row= 82 GeV, then 75.9 GeV < mH< 170 GeV.
The standard model o f electroweak and strong interactions is healthier than it ever was [ 1 ]. However, m a n y problems still remain open. Among them the origin o f the Higgs mechanism and the mysterious Higgs particle attract much attention, ls the Higgs particle really an elementary particle or rather a composite one? Where are the Higgs and top quarks? It is speculated [2] that the most important goal o f experimental particle physics in the next decade is the clarification o f the electroweak symmetry breaking. The solution must be within the TeV energy region. If an experiment can detect a Higgs of 1-2 TeV, then either (a) the Higgs is found; and/or (b) new physics is found; or (c) at least, the onset of the new regime of weak interactions becoming strong can be studied. More quantitatively, the Higgs mass mH is classified into four regions [3]: (i) 10 G e V < m H < 150 GeV, no problems for the standard model up to energies > M p ( ~ 1019 GeV); (ii) 150 G e V < m H < 6 0 0 GeV, information about new physics at a scale below Mp is possible; (iii) mH > 600 GeV, new physics around or below 1 TeV, the standard model description breaks down; (iv) mH < 10 GeV, interesting information about dilatation symmetry is implied. Experimentally, there are almost no reliable lower limits on the Higgs mass [2], while theoretically, there exist some estimations of the lower and upper bounds on mH. The lower bound is constrained by the v a c u u m stability which is usually analyzed in terms of an effective potential calculated to the one-loop level, including both the contributions o f bosons and fermions. On the other hand, the upper bound on mH, mH, is based on requiring the perturbation regime to be valid up to some large energy scale A [ 2 - 9 ]. The value of tnH decreases with the increase of A. Being a physical cut-off, A also appears in the so-called "triviality" problem of the real j.~4 model [ 10-12 ]. It was pointed out that for a stability analysis the A dependence is inevitable whereas the perturbative theory may lose its way. Thus, a variational method in q u a n t u m field theory, the gaussian effective potential ( G E P ) approach, attracts much attention in recent years [ 10-14]. In a previous paper [ 14], we discussed in detail the triviality problem of the ,~4 model. It is shown that after quantization by path integration, the potential in the lagrangian, J On leave of absence from Physics Department, Fudan University, Shanghai, P.R. China. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
161
Volume 200, number 1,2
PHYSICS LETTERS B
V(O) = - ½ao~}2 + (1/4!)2o~} 4 ,
7 January 1988 (1)
evolves into an effective potential U ( ~ ) which has a stable minimum, i.e., a broken phase so long as the following criteria are satisfied: 0 < 2 o <4/J2(#/A) ,
ao > qcrAz ,
(2,3)
where
J2(/2/A) = (1/2re 2) {ln[A//x+ ,/(A/#) 2 + 1] - ( A / / 2 ) / ~
+ 1},
~/c~= ( Z c / ~ c 2 + 1 ) 2o/16n 2 <2o/16re 2 ,
(4) (5)
with Zc = ½exp(8n2/2o + 1 ) >> 1 ,
(6)
and A being a cut-off o f the three-dimensional m o m e n t u m . In this letter the GEP method will be used to discuss the S U ( 2 ) gauge theory coupled to an isospinor of complex scalar fields. The Higgs mechanism in this model enables us to obtain the lower and upper bounds for the mass of Higgs particle to be 0.925mw < mH < 2.07row,
(7)
mw being the mass of the charged vector boson. Beginning from the lagrangian density
£P(x) = - ¼G~a"G,,~,~+ (D~,¢) + (D~'q$) + ao~ + q$- (1/3!)2o(~ + ¢~)2,
(8)
with P v ,u /t v G~/ t / ) =0 ¢t Aa-O A~-g%bcAbAc
,
DF,~=[O~+iA~,~(x)½r~]O(x) ,
(9,10)
and \02/'
0+ = (¢}*0") '
(ll)
we redefine
W~=(I/x/2)(Ag +iAS) , Z~=A~
(12,13)
to represent the charged and neutral massive bosons, respectively. But later on for convenience in calculation, sometimes we shall still use the real A ~ and A2. Also, through a gauge transformation, one can always simplify the isospinor (11 ) as = ~
'(7)
,
(14)
where ~ being a real scalar field with the other three degrees of freedom in (11 ) being absorbed into the Iongitudinal polarizations o f massive W and Z bosons. We write down the hamiltonian of model 8) as usual: H = J - d3x[ ~(H~ 1 2 +H~, +HL
(15)
+H~) + Vl,
where
H¢=OL,e/O~=~, II~8=05HOh,=-h ', 162
(BZ=A~,A~,Z').
(16,17,18)
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
In (15) we have chosen the temporal gauge B ° = 0 so that H 2, =H;A,H;A. etc., and V= 1 (0i~)2 "t- 1 [( OiAij)2 'b ( OiA2j) 2 --O;A~ Oj A i l - O;A~OjA2;] + ½[ (0;Z:) 2 -OiZjOJZ;] -gZz(AzjA ~-Alj A~)
+ g2 Z;Z; W j* W j - g2 Z;Zj W i* W; + gZiJ(A2,A 1;-A2jA~ i) + lg 2(A ~AJi -AJ2A ~)(Az;A lj -AzjA,i) _ ½ao~2 + (1/4!)20~, - ¼g2 ~ Wi~2 _ ~gZ'~O;~- ~g2ZiZ,~2
(A'{ = O;A~ - OJA ~, Z ° = O;Zj - OJZ ', etc.) .
(19)
As in ref. [ 14 ], we quantize this model by evaluating the expectation value of H in a gaussian wave functional (ansatz):
~ = N e x p ( i f P¢,+i f pimlA] +i f P!42A~+i f P;zZ ' x
~:
X
x
_2
1
1f -'0 '
_1
f x y
f [A~(x)-Ai(x)]Fx,,(ff/)[A'l(y)-ff.~(y)]-~ ,f
xy
E=(TIHI~P)
[Ai(x)-Ai(x)lf~v(ff')[Ai(y)-21(y)l
ALl2
)
(20)
= f# x
= f ( ¼f,-.,-(~)+ ~b~,-x(2) + 31~.,.(W) --~If. Ox,,V2f~,l(~)_l - z f ~ x ,-. V ~ f --Tl(7~)_ f,~~. . .v.2.r._.l ( ~v,/ ~ .v
y
12
)
y
+ g2 Z Z ~ ff/j -g2 2,2j ~ ~ + g2[ 3Fz, I ( ITV) F z,J ( Z ) + Z;Z, F z~I ( ff/) + W~;W, F z~' ( 2 ) ] + lg2( ~ Wj - ~ W;)( ~ ~ - ~ IZV,)+g2Fz,) (liE) ~ UVi+ 3g2[Fz,~ ( I~)] 2 - ½ao [(2 + ½F~' (()] + (1/4!)2o {¢4 + 3Fz,.I ( ( ) ~ 2 + ] [Fz~' (()1 z }
_1_~g2[ ~/i[/~ ' -t- ~F ~,1( W)] [~2 -t- ½F2~' (~)1 + Ig2[ 2;2, + 3F.,7~' ( 2 ) 1 [ ¢ 2 + i F L ' ( ( ) ] )
J
(21 )
Then the variational procedures as in ref. [ 14] lead to F~12(l]) = C 3 jf ~ d3p x/p2-t-## 2 exp[iCBp-(x--y)] ,
(22)
d3p F~l(]~) = C 3 f (2z0 3~
(23)
1
exp[iCBp-(x--Y)l,
with ( B = W , Z, ~), Cw=Cz=x/3/-2=C ~/3, C¢= 1, fyF~12(B)FTfl (13)=~ .... The mass parameters .//4 n can be determined by minimizing the energy density # of the system
OglO~'~ I.,4 =;,-~= 0 , ]2~ = _ 0.o _t_~2o[~ I ~ + ~lI i ( / t ~ ) l + ½ g ~[3C11(]A2).. F I~,W,] +~g , 2 [sCII(I~)+ 3 Z;Z;]
(24)
/x2 = 2g 2 CI~ (It 2) + 4g2 I~ lfz; + ~g2[ ~ + ½I. (U ~)],
(25)
#2 =g2 CI, (¢t~) + ~gZ 2,Z, + ~g2( ~ W;) + lgZ [~: + ½11(/~)] +g2 CI. (#~v) •
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Volume 200, number 1,2
PHYSICS LETTERSB
7 January 1988
Notice, however,/* 2 (B = W, Z, ~) is not the mass squared (m 2) of particle B, the latter should be the curvature of the effective potential at the stable broken vacuum. It is easy to prove that the broken vacuum is located at - / Z- t v a c~ - Wvac =0,
(27)
1 ~ac = (6/2o) [ ao -- Z).oI, (U~2 Ivac) -- ]g2CI~(/*2lvac)-]Cg2I,(/*~lva~)]=(3/2o)/*~lva~=_(3/)~o)/*~o,
(28)
while the mass squared of the observable particles are
m 2 w = f [ C I l ( m 2 w ) + C I ~ ( m ~ ) + ~ I ~ (/*¢o2)+ (3/42o)/*~o] ,
(29)
m 2 7 = f [ 2 C i l ( m ~ v ) + ~I~ ~ (/*¢o) 2 + (3/42o)/*~o],
(30)
and m2 =~v2ac().o + b \3 a
I2(/*~o)(J.oa+b) 2 "] a{a[8+2oI2(~o)]+bI2(/*~o)}]'
(31)
where
a = 2 + Cg212 ( m ~v) - Ce gH2 ( m 2 ) I2 ( m~ ) , 3 2 3 2 b= ~g C[ ag C I 2 ( m z2 ) I z ( m w2 ) - I 2 ( m w2 ) - 1/z(m2)] •
It is interesting to see that the mass squared of the W boson or Z boson coincides with/,2 Lva~ or f z I.... but this is not true for the Higgs particle. In all above expressions the integral I~ (/*2) reads: d3p 1 I, (/*~) = f (2g)3 X / ~ - 5 - ~ =U2JI (/*rflA)
=/*~(1/4~2){(A//*~)x/(A/my + 1 --ln [A//*. +,J(A//*~) 2 + 1]}.
(32)
We are not in a position to search for some relations among mw, mz and m . . First, the difference between (29) and (30) can be recast into the following form:
m 2 - m 2 = C f [I, (m~) - I , ( m 2 ) ] - (l/4n 2) Cg2[ m 2 ln(2A/mw) - m27 ln(2A/mz)] = ½Cg2[
2 2 mwI2( mw) - m z l22 ( mz) ] ,
(33)
where the approximation
ln( 2A/mw) =ln( 2A/mz) >> 1
(34)
has been made while eq. (32) and 12=.12 with (4) have beenused. Furthermore, the right-hand side of (33) can be written as
( l/4r~2)Cg2( m~v - m ~ ) ln( 2 A / m , ) .
(35)
If we assume m w = 8 2 GeV, m z = m w ( 1 + e ) , e>0, then m , = m w [ 1 +O(~)1 •
(36)
Hence, the same approximation as in (34) I2(m 2 ) -~I2(m~) = Ie(/*~o) = I 2 ( m ,2 ) ,
(37)
leads to
Cg212(m2,) = 2 . 164
(38)
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Substitution of (37) with (38) into the expression of mH, eq. (31) yields
02 ~/0~ 2 Ivac= m ~ --~] f (~ac(Ao/f - 3.22)(0.642--2o/f)/(7.16+ 2 o / f ) ,
(39)
so the criterion for stability of the broken phase implies 0.642 <2o/g 2 < 3.22.
(40)
On the other hand, there is a well-known relation between mH and mw at the tree level [see eq. (19)] [8,9]
m21m2w = ~2olf .
(41)
A combination of (40) with (41) leads to 0.925 < mH/rnw < 2.07,
(42)
which accomplishes the proof of (7). If m w = 82 GeV, one obtains the lower and upper bounds for the mass of the Higgs particle: 75.9 G e V < mrj < 170 GeV.
(43)
It is interesting to see that our upper bound is not far from that of refs. [9,7], where f f t H = ( l / x f 2 ) ( 3 +
, ~ ) ~/2mw~ 135 GeV and r~H = 125 GeV are given, respectively. On the other hand, the lower bound in (43) seems much higher than that in ref. [4]. Although a series of difficulties has to be overcome before one can manipulate a realistic S U ( 2 ) × U ( I ) model and take the effect of fermions into account, our treatment does share the advantages of unity in point of view. We treat both the lower and upper bounds by the vacuum stability condition as shown in eq. (39). Besides (20), only a further approximation (33) or (34) is made. Eqs. ( 3 6 ) - ( 3 8 ) imply that the accuracy of the estimation would not be too bad so long as the cut-offA is large and the coupling constant g2 is small. Since the Yukawa coupling is even weaker than g2, we expect that the fermions will play a minor role in enhancing the instability of the vacuum till the mass of the fermion becomes very heavy. Correspondingly, we tentatively combine our lower-upper bounds with the plot in ref. [ 7] and are able to estimate the mass of the top quark: rnt> 125 GeV.
(44)
As recent experiments, especially the measurements on B°-13° mixing, imply that the top quark has a large mass [1,15,16], the estimated value in (44) may be also interesting. Hopefully, both (43) and (44) could serve as some test of the standard model. This work is partially supported by the NSF of China under Grant NO.KR12040. One of the authors (GuangJiong Ni) has benefitted from the lectures given at the Conference on Phenomenology in High Energy Physics (20-22 July 1987) and the Meeting on Search for Scalar Particles: Experimental and Theoretical Aspects (23, 24 July 1987) held at the International Centre for Theoretical Physics, Trieste, where this work was completed. He would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References [ l ] P. Darriulat, Lectureat Europian Intern. High energyphysicsConf. (Uppsala, Sweden, 25 June-1 July 1987). [2 ] G. Altarelli, Lectureat Meetingon Search for scalar particles: experimental and theoretical aspects (Trieste, Italy, 23, 24 July 1987). [3] C. Wetterich, Lecture at Meeting on Search for scalar particles: experimental and theoretical aspects (Trieste, Italy, 23, 24 July 1987). [4] N. Cabibbo, L. Maiani, G. Praisi and R. Petronzio, Nucl. Phys. B 158 (1979) 295. [5] L Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B 136 (1978) 115. 165
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[6] R. Dashen and H. Neuberger, Phys. Rev. Lett. 50 (1983) 1897. [7] M.A.B. Beg, C. Panagiotakopoulos and A. Sirlin, Phys. Rev. Left. 52 (1984) 883. [8] D.J.E. Callaway, Nucl. Phys. B 233 (1984) 189. [9] D.J.E. Callaway and R. Petronzio, Nucl. Phys. B 240 (1984) 577; B 267 (1986) 253; B 277 (1986) 50. [ 10] P.M. Stevenson, B. Alles and R. Tarrch, Phys. Rev. D 35 (1987) 2407, and references therein. [ 11 ] M. Consoli and G. Passarino, Phys. Lett. B 165 (1985) 113. [ 12] Y. Brihaye and M. Consoli, Nuovo Cimento 94A (1986) 1. [ 13] R. Ingermanson, Nucl. Phys. B 266 (1986) 620. [ 14] G,J. Ni, S.Y. Lou and S.Q. Chen, Fudan University preprint (1987), [ 15] A. Ali, Lecture at Conf. on Phenomenology in high energy physics (Trieste, Italy, 20-22 June 1987). [ 16] D,S. Du and Z.Y. Zhao, Princeton University preprint IASSNS-HEP-87/23 (April 1987).
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