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Bounds on the top-quark mass in N=1 supergravity models
Volume 193, number I
PHYSICS LETTERS B
9 July 1987
B O U N D S O N THE TOP-QUARK MASS IN N = 1 SUPERGRAVITY MODELS
C. CHIOU-LAHANAS, B.C. GEORGALAS and C.G. PAPADOPOULOS Nuclear and Particle Physics Section, Universityof Athens, Athens 15 771, Greece Received 18 March 1987
From the tree level study of the SU (3) X SU (2) X U (1) model arising as the flat limit of a spontaneously broken N = 1 supergravity, via the super Higgs mechanism, we find particular directions of minima which lead to upper bound for the top-quark mass. These bounds on m t can also be of relevance to superstring-inspired low-energy models.
In the S U ( 3 ) x S U ( 2 ) × U ( 1 ) supergravity models [1] at, where the symmetry breaking SU(3) x SU(2) xU(1 ) ~ S U ( 3 ) XUem(1 ) is obtained through radiative effects [ 3 ], the tree level study of the effective potential at the unification scale (Mx) leads to relations between the parameters of the scalar potential in order that the SU (3) X SU (2) × U ( 1) be the absolute minimum of the theory leaving therefore the theory unbroken. This demand provides useful relations since other possible minima of the theory should lie higher than the unbroken SU(3)SU(2)U(1 ) minimum, restricting therefore the allowed domain of the free parameters of the theory. This may have severe consequences for the phenomenology of these models. However in all models considered so far the minima were sought in directions in which the D-terms either vanished or had negligible contribution [4]. These considerations leave out the possible existence of other symmetry breaking minima in which D-terms are not zero, therefore not taking into account other bounds which may exist and are perhaps more restrictive. In this letter we consider minima for which the D-terms do not vanish and show that interesting relations among the parameters of the theory (soft SUSY breaking terms, Yukawa coupling) should be fulfilled by demanding an S U ( 3 ) x S U ( 2 ) x U ( I ) symmetric theory at energies E~-Mp. One of these relations which is particularly studied here provides an upper bound for the top-quark mass. The superpotential of the SU(3)X SU(2)X U(1 ) theory under consideration is assumed to have the wellknown form [ 3 ]
f= ht EijH2iQaj VCa+ hb EijHu OajDCa+ fL H1, ¢~jLjE~+ m4 EijH2iH U ,
( 1)
where (ht, hb, f) are taken to be real constants and co, is the total antisymmetric symbol in two dimensions
(~12= l). The physical content of the S U ( 3 ) / S U (2) / U (I ) theory is given in table I #2. Since the Yukawa couplings hb and fL are much smaller than h~ (Yukawa coupling of the heaviest quark) we can safely ignore them, so that ftakes the form
f = ht EoH2iQaj VCa+ m4 EoH2iH u •
(1 ')
We shall see from the following tree level analysis that the choice (hb=fm=0) does not affect the results. As ~' Fora reviewsee for instanceref. [2]. ~2 The superpotential (1) involves ordinary matter and Higgs fields.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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9 July 1987
PHYSICS LETTERS B
Volume 193, number 1 Table 1 Chiral multiplets
Transforms under SU(3) × SU(2) × U ( I )
.Q V~ D¢ £ E¢ //2 HI
(3, 2, 1/6) (3, 1, - 2 / 3 ) (3, l, 1/3) (1, 2, - 1/2)
(I, I, I) (1, 2, 1/2) (1, 2, - 1/2)
is well known at scales ~Mp which is assumed to be the flat limit of the broken N= 1 supergravity (SUGRA) the effective potential has a form [ 1,2 ]
Of ~.2 Veff=~]-~+m3/2Z +(A-3)m3/2(f+f*)+D-terms.
(2)
With the field content of (1') we have
Vefr=m2(HiH{ + H 2 H f )+mZ/~(HIH~ + H 2 H f +QQ+ V+ Ve) + ( A - 1 )m3/2m4(~.H2Hi +h.c.) +htm4(QVcH + +h.c.) +Ahtm3/2(~H2QVc +h.c.) + h 2 [ ( H 2 H f )(Vc+ Vc)+(QV)(V+ Q+ ) + (H2Q)(Q * H?~+ )] +D-terms,
(3)
where the D-terms are explicitly given by
D~=½gZ[~;(QQ+ ) _ ] ( v + Vc)+ ½(H2Hf ) - ½ ( H I H + )] 2 , 2 I 2 +)2 +½(H2H~- )2 + ½ ( Q Q + ) 2 - ( H I H + ) ( H z H 2 ) - ( D2=4g2[½(H~HI
(4a) H 2H 2+) ( Q Q + )
- (QQ + )(H~ H~- ) + 2 (H2H ~-)(H, H~- ) + 2(QH~- )(HI Q + ) + 2 ( Q H ~ )(HE Q + ) ] , D 2 = ]g][ ] (QQ + )2 + ~ ( V + V¢)Z + ~(QQ+)( v + v¢) - 2 (QV¢)( V+ Q+ ) ] .
(4b) (4c)
Searching for minima of Vefr we can choose, without loss of generality, particular directions due to the SU(3) × S U ( 2 ) x U ( 1 ) invariance of Veff, so that,
Q,,j=83~2;IQIe iq . For the remaining fields we use the following parametrization: V== IIVll(e iv~ sin 0 cos ~, e i~ sin 0 sin 0, ei~ cos 0) , H l V = IIHI II(e ih'' c o s x i ,
e ih'2 sin xl) ,
H2V= IIn21l(e ih~' cosx2, e ih:: sin x2) ,
where IlZ/ll2=~j IZf[ 2 and 0~<0, ¢~, Xl, x2<~n/2. Rescaling the fields (Zr-,m3/zZ~) and using the above parametrization Vefrcan be cast into the form Veer=m ,I3/217~fr, with P dimensionless, as given by the following expression:
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Volume 193, number I
9 July 1987
PHYSICS LETTERS B
Veer= m~/2 {(1 +g2)( IIH, II2 + IIH: II2) +
(I
QI 2 + UI012)
+ 2 ( A - 1 )#IIHI IIliB: II[sin x~ cos x: cos(h~2 +h21 ) - sin x: cos x~ cos(h~ + h2:)] + 2#ht IOl UH~ IIII10lcos 0 sin x~ cos(q+ v3 - h t 2 ) +2Aht I QI UH2 IIIIVllcos 0 cos x2 cos(q+ 03 +hz~ ) + h,2( liB2 II2 IIV]l2 + IIVll21QI %0s20 + I QI: liB: U:cos2xz) } +D-terms,
(5)
where/z--m4/m3/2. The Dl-term is as given in eq. (4a), while D2 and D3 obtain the form D~ = ]g~[( I QI 2 _ IIn~ II2cos 2x~ - liB2 IIZcos 2x2) 2 + (lln~ II:sin 2x~ + liB: II2sin 2x2) 2 -41ln~ II2 liB2 II:sin 2x~ sin 2x2 cos(hi2 +hzl - h i , -h22)] ,
(5a)
D~ = ~g][( I QI 2 + IIVll 2)2 _ 411Vll21QI cos20] .
(5b)
Looking for minima in directions in which the D-terms vanish we distinguish the following interesting cases: Case (A) n~=0,
x2=0=0
and
IIQIl=llI'ql=lln211=x,
Case (B)
Q= V=0, Hit = H 2 2 = 0
and
H?12= + H 2 ~ - H .
In case (A) the Verfbecomes
Veff=m'~/2[(3+lz2)xa -1"2ahtX3 + 3ht2x4] ,
x~>0,
so that a neccessary and sufficient condition for not having a minimum with x ~ 0 lying lower than the symmetric one is the well-known relation [3,5] (see fig. 1 ) A2---<3(3+/z 2)
(6)
•
In the second ease (B) Vefr= 2m4/2[(1 + / ? ) + ( A - 1 )/z] IHI 2 , and the stability condition at ~ M p provides the bound I A - I I ~< lul + 1/I/~1,
(7)
which is equivalent to the known condition ml2 + m22> 21rn~
(see refs. [ 3,5]).
o
"%~t;~. ,#a
" ° ~z¢'<
"'*,n,- q g . , .
"%,
"%. \ I ' -10
i -II
i -S
~ -4
l
l -Z
4
6
I $
10
A Fig. 1. The dashed line is the curve A 2= 3 ( 3 +/z 2). The region A 2> 3 ( 3 +/z 2) in the (A,/z ) plane is the region of false color minima ( case (A)). The solid line is the curve IA - 11 = I# [ + 1/ [/zl. The region IA - 1 [ > [# I + 1/ i# [ is the region where the potential becomes unbounded from below (case (B)).
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Volume 193, number 1
PHYSICS LETTERSB
9 July 1987
We consider now the particular direction for which the D-terms ~ 0, 0 =x2 = 0 and x~ = n/2. This can be seen to be a solution of the minimization equations for the Verrand hence this minimum should be taken into account. In this direction V~frbecomes
V~=m']/2 [(1 +/x2)( IIH, IIz + 11/-/2II~) + ( I QI 2 + IIvii 2) - 2 I A - I I I/tl HH, IIIIHz [Icos(h2, + h ~ 2 ) - 2 IPlht I QI I[Vl{I[Hl Ilcos(q+v3 +h~2)
- 2 IAlht I QI IIVIIIIH~IIcos(q+ v3 + h2, ) + h,Z( IIH2 II2 IIVIi2 + IIVIi2 I QI 2 + i Qi 2 IIH2112)]+D-terms,
(8)
where a 2 = ½g21(~ [Q[ 2 _ ][I VII2 + ½[[H2 [[2 _
2)2,
½ [IH~ [[
a 2 = ~g2( ]Q[ 2 + [[H~ l[2 _ [[H2[[2 ) 2 ,
D32 =~g]([QI2-[]VI[2)2. Without loss of generality we a s s u m e ht > 0. Also, being at scales Mp we set ~ g~ =g2 =g3 =g'. When the ratio g'/ht is not large enough, undesired minima, lying lower than the symmetric one, appear for the values of A and g in the allowed region coming from conditions (6), (7). To avoid these minima a condition of the form
g' /2h, >~Jo( A, /.t) must be satisfied. This condition is equivalent to Veff>~0 preventing therefore the appearance of a minimum lying lower than the symmetric one. The function Jo(A, p) which appears in the inequality above is a function of the parameters A and/a whose explicit form we do not give. The condition above provides the following upper bound for the Yukawa coupling ht at Mx ht I~, <~g'/2Jo(A, lt) .
(9)
The minima considered so far are also local minima in the case of nonvanishing Yukawa couplings hb, fL, provided that the fields E, L, D have vanishing VEV's. The evolution of the Yuakwa coupling ht at any scale is given by the expression [ 3,5,6]
hZ,(t)=h2, fg~E(t)/[l+(3/8n2)hZt [MxF(t)] ,
t-lnMZx/Q z
(10)
where E(t), F(t) are positive functions of t depending on the running coupling constants only. As one can see the expression (10) is a monotonically increasing function of hZtMx. From this expression and the following relation [ 3 ]:
mE
hZt(Q) 0
mE--~z---~=
M,
w,
(ll)
The above bound of the Yukawa coupling becomes an upper bound for the top-quark mass for given values of A and/1. Alternatively by fixing the upper bound (mr)max for rnt the allowed values of A and/t parameters are restricted to be within a certain domain. In fig. 2 we have drawn the allowed region in the (A,/t) plane for (mr)max=30 GeV and (mt)m~x= 100 GeV. Notice that the (m0m~x=30 GeV and (mt)m~x=100 GeV boundaries are very close to each other. Any point in the excluded region in fig. 2 gives a tree level breaking of the SU(3) × SU(2) × U(1 ) to minima breaking color and electromagnetism. For comparison we have also drawn the A,/z domain allowed when the conditions (6), (7) are imposed, showing that the bounds stemming from the minima we have considered are more restrictive. We observe that for values of A in the range - 1 < A < 2 the local minima we have investigated lie higher than the symmetric one for any value of ht providing no upper bound for rot. The SU(3) × SU(2) X U(1 ) symmetry should remain unbroken not only at large scales but also at any scale 58
Volume 193, number 1
PHYSICS LETTERS B
9 July 1987
, / (rnt)mox 100 G~v
X'X'N\
\\
\(mt)max=lO0 Ge~
///////
///
/
'
//
/// unstable minimum \\, I
I
I
-.1
-2
-I
2
minimum
3
&
A
Fig. 2. Permitted A,/~ values for a given maximum tdp 0 (A < 0).
Q in the region/14, < Q < Mp. At scales lower than the Planck scale and with the parametrization (5) the Veff is written as a function of running parameters.
f'(t) =m~/2trh2QlQI2+ rh2vI VI 2 +rhl2 In~ 12 + m ] I//212 + 2Btrh4 In111/-/2 + 2Atht I VI IQI I//2 Icosxz+2h, m41 VI [QI IH~ Isin x I +ht( IQI21 v I 2 + +D-terms.
Isin(x~ --x2) I vI 2 Ia212 + I//212 IQI2)] (12)
The running parameters of the above expression are the solutions of the renormalization group equations which have been taken from ref. [ 7 ] with the following initial Conditions: rh2(gx)=# 2 ,
At(gx)=A,
Bt(gx)=A-1,
rh2(Mx) =rh2(Mx)= 1 +/z 2 , rh~=rh 2 = 1 , rh,=mi/m3/2
•
We have minimized the f'(t) (eq. (12)) numerically at any scale t. By the requirement that SU ( 3 ) × S U ( 2 ) × U(I ) vacuum remains stable against tunneling into the fals vacua we considered here, that is the vacua in the direction x~ = n/2, Z2 = 0 = 0, we get an upper bound for the running Yukawa coupling of the form ht(t) ~
t),
which results to an upper bound for the top-quark mass. The bounds obtained for Q < Mx are more stringent than those obtained for Q=Mx. The results are shown in fig. 3, where this upper bound for mt, (rot) . . . . is drawn as function of t for various values of A and #. Choosing for instance A = 3, # = 3 we see that in order for SU( 3 ) × S U ( 2 ) × U(I ) to remain unbroken at M~, m~ must be less than 95 GeV but this bound becomes stronger, mr< 55 GeV, when the same stability condition is imposed at scales Mw that is (mr)max is reduced from its value at Mx by about 40%. Reductions of this size to (mr) maxas one moves from Mx to Mw are obtained for the other combinations of A and/z. The above analysis can be carded over almost intact to the superstring inspired models in which the group E6 breaks down to S U ( 3 ) × S U ( 2 ) × U ( 1 ) × U ' ( 1 ) [8] by the Hosotani mechanism [9]. In these models the superpotential has the form
f= ht EijH2iQ=j~ + 2~ijNH2iHv + ....
(13) 59
Volume 193, number I
X 0
PHYSICS LETTERS B
9 July 1987
IO0
E eo E 5o ~,0 ZO , _ _ _ - - - -
0i
110
I ZO
I 30
I 40
I 50
I 60
I 70
t = In PlxZ/QZ
Fig. 3. Upper bound for the top-quark mass versus t-= In M ~ / Q 2 . The solid line corresponds to A = 3,/z = 3, the dashed dotted line corresponds to A = 4 , / t = 4 and the dashed line corresponds to A=-3,#=4.
where N is a singlet u n d e r the S U ( 3 ) X S U ( 2 ) × U ( 1 ) s u b g r o u p b u t it carries the U ' ( 1 ) q u a n t u m n u m b e r s . We can see f r o m (13) that 2 ( N ) plays the role o f m4 in the m o d e l (1'). T h e r e f o r e the analysis regarding the t o p - q u a r k m a s s still holds. A detailed analysis r e g a r d i n g the s u p e r s t r i n g low-energy m o d e l s will be given in a future p u b l i c a t i o n . We wish to t h a n k Professor A.B. L a h a n a s for helpful d i s c u s s i o n s a n d e n c o u r a g e m e n t . We acknowledge the s u p p o r t o f the G r e e k M i n i s t r y o f R e s e a r c h a n d Technology. References
[ 1] J. Ellis and D.V. Nanopoulos, Phys. Lett. B 116 (1982) 133; R. Barbieri, S. Ferrara, D.V. Nanopoulos and K.S. Stelle, Phys. Lett. B 113 (1982) 219; H.P. Nilles, M. Srednicki and D. Wyler, Phys. Lett. B 120 (1982) 346; L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D 27 (1983) 2359. [2] H.P. Nilles, Phys. Rep. C I l0 (1984) 1. [3] J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 121 (1983) 123; L. Ib~ifiezand C. Lopez, Phys. Lett. B 126 (1983) 54; J. Ellis, J. Hagelin, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 125 (1983) 275; C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quiros, Nucl. Phys. B 236 (1984) 438. [4] M. Drees, M. Gliick and K. Grassie, Phys. Lett. B 157 (1985) 164; B 159 (1985) 118. [5] L. Ib~lfiez,C. Lopez and C. Munoz, Nucl. Phys. B 256 (1985) 218. [6] A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. C 145 (1987) 1. [7] K. Inoue, A. Kakuto, H. Komatsu and H. Takeshita, Prog. Theor. Phys. 68 (1982) 927; 71 (1984) 348. [8] J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Mod. Phys. Lett. A l (1986) 57. [9] Y. Hosotani, Phys. Lett. B 126 (1983) 309.
60
PHYSICS LETTERS B
9 July 1987
B O U N D S O N THE TOP-QUARK MASS IN N = 1 SUPERGRAVITY MODELS
C. CHIOU-LAHANAS, B.C. GEORGALAS and C.G. PAPADOPOULOS Nuclear and Particle Physics Section, Universityof Athens, Athens 15 771, Greece Received 18 March 1987
From the tree level study of the SU (3) X SU (2) X U (1) model arising as the flat limit of a spontaneously broken N = 1 supergravity, via the super Higgs mechanism, we find particular directions of minima which lead to upper bound for the top-quark mass. These bounds on m t can also be of relevance to superstring-inspired low-energy models.
In the S U ( 3 ) x S U ( 2 ) × U ( 1 ) supergravity models [1] at, where the symmetry breaking SU(3) x SU(2) xU(1 ) ~ S U ( 3 ) XUem(1 ) is obtained through radiative effects [ 3 ], the tree level study of the effective potential at the unification scale (Mx) leads to relations between the parameters of the scalar potential in order that the SU (3) X SU (2) × U ( 1) be the absolute minimum of the theory leaving therefore the theory unbroken. This demand provides useful relations since other possible minima of the theory should lie higher than the unbroken SU(3)SU(2)U(1 ) minimum, restricting therefore the allowed domain of the free parameters of the theory. This may have severe consequences for the phenomenology of these models. However in all models considered so far the minima were sought in directions in which the D-terms either vanished or had negligible contribution [4]. These considerations leave out the possible existence of other symmetry breaking minima in which D-terms are not zero, therefore not taking into account other bounds which may exist and are perhaps more restrictive. In this letter we consider minima for which the D-terms do not vanish and show that interesting relations among the parameters of the theory (soft SUSY breaking terms, Yukawa coupling) should be fulfilled by demanding an S U ( 3 ) x S U ( 2 ) x U ( I ) symmetric theory at energies E~-Mp. One of these relations which is particularly studied here provides an upper bound for the top-quark mass. The superpotential of the SU(3)X SU(2)X U(1 ) theory under consideration is assumed to have the wellknown form [ 3 ]
f= ht EijH2iQaj VCa+ hb EijHu OajDCa+ fL H1, ¢~jLjE~+ m4 EijH2iH U ,
( 1)
where (ht, hb, f) are taken to be real constants and co, is the total antisymmetric symbol in two dimensions
(~12= l). The physical content of the S U ( 3 ) / S U (2) / U (I ) theory is given in table I #2. Since the Yukawa couplings hb and fL are much smaller than h~ (Yukawa coupling of the heaviest quark) we can safely ignore them, so that ftakes the form
f = ht EoH2iQaj VCa+ m4 EoH2iH u •
(1 ')
We shall see from the following tree level analysis that the choice (hb=fm=0) does not affect the results. As ~' Fora reviewsee for instanceref. [2]. ~2 The superpotential (1) involves ordinary matter and Higgs fields.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
55
9 July 1987
PHYSICS LETTERS B
Volume 193, number 1 Table 1 Chiral multiplets
Transforms under SU(3) × SU(2) × U ( I )
.Q V~ D¢ £ E¢ //2 HI
(3, 2, 1/6) (3, 1, - 2 / 3 ) (3, l, 1/3) (1, 2, - 1/2)
(I, I, I) (1, 2, 1/2) (1, 2, - 1/2)
is well known at scales ~Mp which is assumed to be the flat limit of the broken N= 1 supergravity (SUGRA) the effective potential has a form [ 1,2 ]
Of ~.2 Veff=~]-~+m3/2Z +(A-3)m3/2(f+f*)+D-terms.
(2)
With the field content of (1') we have
Vefr=m2(HiH{ + H 2 H f )+mZ/~(HIH~ + H 2 H f +QQ+ V+ Ve) + ( A - 1 )m3/2m4(~.H2Hi +h.c.) +htm4(QVcH + +h.c.) +Ahtm3/2(~H2QVc +h.c.) + h 2 [ ( H 2 H f )(Vc+ Vc)+(QV)(V+ Q+ ) + (H2Q)(Q * H?~+ )] +D-terms,
(3)
where the D-terms are explicitly given by
D~=½gZ[~;(QQ+ ) _ ] ( v + Vc)+ ½(H2Hf ) - ½ ( H I H + )] 2 , 2 I 2 +)2 +½(H2H~- )2 + ½ ( Q Q + ) 2 - ( H I H + ) ( H z H 2 ) - ( D2=4g2[½(H~HI
(4a) H 2H 2+) ( Q Q + )
- (QQ + )(H~ H~- ) + 2 (H2H ~-)(H, H~- ) + 2(QH~- )(HI Q + ) + 2 ( Q H ~ )(HE Q + ) ] , D 2 = ]g][ ] (QQ + )2 + ~ ( V + V¢)Z + ~(QQ+)( v + v¢) - 2 (QV¢)( V+ Q+ ) ] .
(4b) (4c)
Searching for minima of Vefr we can choose, without loss of generality, particular directions due to the SU(3) × S U ( 2 ) x U ( 1 ) invariance of Veff, so that,
Q,,j=83~2;IQIe iq . For the remaining fields we use the following parametrization: V== IIVll(e iv~ sin 0 cos ~, e i~ sin 0 sin 0, ei~ cos 0) , H l V = IIHI II(e ih'' c o s x i ,
e ih'2 sin xl) ,
H2V= IIn21l(e ih~' cosx2, e ih:: sin x2) ,
where IlZ/ll2=~j IZf[ 2 and 0~<0, ¢~, Xl, x2<~n/2. Rescaling the fields (Zr-,m3/zZ~) and using the above parametrization Vefrcan be cast into the form Veer=m ,I3/217~fr, with P dimensionless, as given by the following expression:
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Volume 193, number I
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PHYSICS LETTERS B
Veer= m~/2 {(1 +g2)( IIH, II2 + IIH: II2) +
(I
QI 2 + UI012)
+ 2 ( A - 1 )#IIHI IIliB: II[sin x~ cos x: cos(h~2 +h21 ) - sin x: cos x~ cos(h~ + h2:)] + 2#ht IOl UH~ IIII10lcos 0 sin x~ cos(q+ v3 - h t 2 ) +2Aht I QI UH2 IIIIVllcos 0 cos x2 cos(q+ 03 +hz~ ) + h,2( liB2 II2 IIV]l2 + IIVll21QI %0s20 + I QI: liB: U:cos2xz) } +D-terms,
(5)
where/z--m4/m3/2. The Dl-term is as given in eq. (4a), while D2 and D3 obtain the form D~ = ]g~[( I QI 2 _ IIn~ II2cos 2x~ - liB2 IIZcos 2x2) 2 + (lln~ II:sin 2x~ + liB: II2sin 2x2) 2 -41ln~ II2 liB2 II:sin 2x~ sin 2x2 cos(hi2 +hzl - h i , -h22)] ,
(5a)
D~ = ~g][( I QI 2 + IIVll 2)2 _ 411Vll21QI cos20] .
(5b)
Looking for minima in directions in which the D-terms vanish we distinguish the following interesting cases: Case (A) n~=0,
x2=0=0
and
IIQIl=llI'ql=lln211=x,
Case (B)
Q= V=0, Hit = H 2 2 = 0
and
H?12= + H 2 ~ - H .
In case (A) the Verfbecomes
Veff=m'~/2[(3+lz2)xa -1"2ahtX3 + 3ht2x4] ,
x~>0,
so that a neccessary and sufficient condition for not having a minimum with x ~ 0 lying lower than the symmetric one is the well-known relation [3,5] (see fig. 1 ) A2---<3(3+/z 2)
(6)
•
In the second ease (B) Vefr= 2m4/2[(1 + / ? ) + ( A - 1 )/z] IHI 2 , and the stability condition at ~ M p provides the bound I A - I I ~< lul + 1/I/~1,
(7)
which is equivalent to the known condition ml2 + m22> 21rn~
(see refs. [ 3,5]).
o
"%~t;~. ,#a
" ° ~z¢'<
"'*,n,- q g . , .
"%,
"%. \ I ' -10
i -II
i -S
~ -4
l
l -Z
4
6
I $
10
A Fig. 1. The dashed line is the curve A 2= 3 ( 3 +/z 2). The region A 2> 3 ( 3 +/z 2) in the (A,/z ) plane is the region of false color minima ( case (A)). The solid line is the curve IA - 11 = I# [ + 1/ [/zl. The region IA - 1 [ > [# I + 1/ i# [ is the region where the potential becomes unbounded from below (case (B)).
57
Volume 193, number 1
PHYSICS LETTERSB
9 July 1987
We consider now the particular direction for which the D-terms ~ 0, 0 =x2 = 0 and x~ = n/2. This can be seen to be a solution of the minimization equations for the Verrand hence this minimum should be taken into account. In this direction V~frbecomes
V~=m']/2 [(1 +/x2)( IIH, IIz + 11/-/2II~) + ( I QI 2 + IIvii 2) - 2 I A - I I I/tl HH, IIIIHz [Icos(h2, + h ~ 2 ) - 2 IPlht I QI I[Vl{I[Hl Ilcos(q+v3 +h~2)
- 2 IAlht I QI IIVIIIIH~IIcos(q+ v3 + h2, ) + h,Z( IIH2 II2 IIVIi2 + IIVIi2 I QI 2 + i Qi 2 IIH2112)]+D-terms,
(8)
where a 2 = ½g21(~ [Q[ 2 _ ][I VII2 + ½[[H2 [[2 _
2)2,
½ [IH~ [[
a 2 = ~g2( ]Q[ 2 + [[H~ l[2 _ [[H2[[2 ) 2 ,
D32 =~g]([QI2-[]VI[2)2. Without loss of generality we a s s u m e ht > 0. Also, being at scales Mp we set ~ g~ =g2 =g3 =g'. When the ratio g'/ht is not large enough, undesired minima, lying lower than the symmetric one, appear for the values of A and g in the allowed region coming from conditions (6), (7). To avoid these minima a condition of the form
g' /2h, >~Jo( A, /.t) must be satisfied. This condition is equivalent to Veff>~0 preventing therefore the appearance of a minimum lying lower than the symmetric one. The function Jo(A, p) which appears in the inequality above is a function of the parameters A and/a whose explicit form we do not give. The condition above provides the following upper bound for the Yukawa coupling ht at Mx ht I~, <~g'/2Jo(A, lt) .
(9)
The minima considered so far are also local minima in the case of nonvanishing Yukawa couplings hb, fL, provided that the fields E, L, D have vanishing VEV's. The evolution of the Yuakwa coupling ht at any scale is given by the expression [ 3,5,6]
hZ,(t)=h2, fg~E(t)/[l+(3/8n2)hZt [MxF(t)] ,
t-lnMZx/Q z
(10)
where E(t), F(t) are positive functions of t depending on the running coupling constants only. As one can see the expression (10) is a monotonically increasing function of hZtMx. From this expression and the following relation [ 3 ]:
mE
hZt(Q) 0
mE--~z---~=
M,
w,
(ll)
The above bound of the Yukawa coupling becomes an upper bound for the top-quark mass for given values of A and/1. Alternatively by fixing the upper bound (mr)max for rnt the allowed values of A and/t parameters are restricted to be within a certain domain. In fig. 2 we have drawn the allowed region in the (A,/t) plane for (mr)max=30 GeV and (mt)m~x= 100 GeV. Notice that the (m0m~x=30 GeV and (mt)m~x=100 GeV boundaries are very close to each other. Any point in the excluded region in fig. 2 gives a tree level breaking of the SU(3) × SU(2) × U(1 ) to minima breaking color and electromagnetism. For comparison we have also drawn the A,/z domain allowed when the conditions (6), (7) are imposed, showing that the bounds stemming from the minima we have considered are more restrictive. We observe that for values of A in the range - 1 < A < 2 the local minima we have investigated lie higher than the symmetric one for any value of ht providing no upper bound for rot. The SU(3) × SU(2) X U(1 ) symmetry should remain unbroken not only at large scales but also at any scale 58
Volume 193, number 1
PHYSICS LETTERS B
9 July 1987
, / (rnt)mox 100 G~v
X'X'N\
\\
\(mt)max=lO0 Ge~
///////
///
/
'
//
/// unstable minimum \\, I
I
I
-.1
-2
-I
2
minimum
3
&
A
Fig. 2. Permitted A,/~ values for a given maximum tdp
Q in the region/14, < Q < Mp. At scales lower than the Planck scale and with the parametrization (5) the Veff is written as a function of running parameters.
f'(t) =m~/2trh2QlQI2+ rh2vI VI 2 +rhl2 In~ 12 + m ] I//212 + 2Btrh4 In111/-/2 + 2Atht I VI IQI I//2 Icosxz+2h, m41 VI [QI IH~ Isin x I +ht( IQI21 v I 2 + +D-terms.
Isin(x~ --x2) I vI 2 Ia212 + I//212 IQI2)] (12)
The running parameters of the above expression are the solutions of the renormalization group equations which have been taken from ref. [ 7 ] with the following initial Conditions: rh2(gx)=# 2 ,
At(gx)=A,
Bt(gx)=A-1,
rh2(Mx) =rh2(Mx)= 1 +/z 2 , rh~=rh 2 = 1 , rh,=mi/m3/2
•
We have minimized the f'(t) (eq. (12)) numerically at any scale t. By the requirement that SU ( 3 ) × S U ( 2 ) × U(I ) vacuum remains stable against tunneling into the fals vacua we considered here, that is the vacua in the direction x~ = n/2, Z2 = 0 = 0, we get an upper bound for the running Yukawa coupling of the form ht(t) ~
t),
which results to an upper bound for the top-quark mass. The bounds obtained for Q < Mx are more stringent than those obtained for Q=Mx. The results are shown in fig. 3, where this upper bound for mt, (rot) . . . . is drawn as function of t for various values of A and #. Choosing for instance A = 3, # = 3 we see that in order for SU( 3 ) × S U ( 2 ) × U(I ) to remain unbroken at M~, m~ must be less than 95 GeV but this bound becomes stronger, mr< 55 GeV, when the same stability condition is imposed at scales Mw that is (mr)max is reduced from its value at Mx by about 40%. Reductions of this size to (mr) maxas one moves from Mx to Mw are obtained for the other combinations of A and/z. The above analysis can be carded over almost intact to the superstring inspired models in which the group E6 breaks down to S U ( 3 ) × S U ( 2 ) × U ( 1 ) × U ' ( 1 ) [8] by the Hosotani mechanism [9]. In these models the superpotential has the form
f= ht EijH2iQ=j~ + 2~ijNH2iHv + ....
(13) 59
Volume 193, number I
X 0
PHYSICS LETTERS B
9 July 1987
IO0
E eo E 5o ~,0 ZO , _ _ _ - - - -
0i
110
I ZO
I 30
I 40
I 50
I 60
I 70
t = In PlxZ/QZ
Fig. 3. Upper bound for the top-quark mass versus t-= In M ~ / Q 2 . The solid line corresponds to A = 3,/z = 3, the dashed dotted line corresponds to A = 4 , / t = 4 and the dashed line corresponds to A=-3,#=4.
where N is a singlet u n d e r the S U ( 3 ) X S U ( 2 ) × U ( 1 ) s u b g r o u p b u t it carries the U ' ( 1 ) q u a n t u m n u m b e r s . We can see f r o m (13) that 2 ( N ) plays the role o f m4 in the m o d e l (1'). T h e r e f o r e the analysis regarding the t o p - q u a r k m a s s still holds. A detailed analysis r e g a r d i n g the s u p e r s t r i n g low-energy m o d e l s will be given in a future p u b l i c a t i o n . We wish to t h a n k Professor A.B. L a h a n a s for helpful d i s c u s s i o n s a n d e n c o u r a g e m e n t . We acknowledge the s u p p o r t o f the G r e e k M i n i s t r y o f R e s e a r c h a n d Technology. References
[ 1] J. Ellis and D.V. Nanopoulos, Phys. Lett. B 116 (1982) 133; R. Barbieri, S. Ferrara, D.V. Nanopoulos and K.S. Stelle, Phys. Lett. B 113 (1982) 219; H.P. Nilles, M. Srednicki and D. Wyler, Phys. Lett. B 120 (1982) 346; L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D 27 (1983) 2359. [2] H.P. Nilles, Phys. Rep. C I l0 (1984) 1. [3] J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 121 (1983) 123; L. Ib~ifiezand C. Lopez, Phys. Lett. B 126 (1983) 54; J. Ellis, J. Hagelin, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 125 (1983) 275; C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quiros, Nucl. Phys. B 236 (1984) 438. [4] M. Drees, M. Gliick and K. Grassie, Phys. Lett. B 157 (1985) 164; B 159 (1985) 118. [5] L. Ib~lfiez,C. Lopez and C. Munoz, Nucl. Phys. B 256 (1985) 218. [6] A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. C 145 (1987) 1. [7] K. Inoue, A. Kakuto, H. Komatsu and H. Takeshita, Prog. Theor. Phys. 68 (1982) 927; 71 (1984) 348. [8] J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Mod. Phys. Lett. A l (1986) 57. [9] Y. Hosotani, Phys. Lett. B 126 (1983) 309.
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