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Bounds on masses of Higgs boson and heavy fermions in a softly broken supersymmetric model
Volume 129, number 1,2
PHYSICS LETTERS
15 September 1983
BOUNDS ON MASSES OF HIGGS BOSON AND HEAVY FEILMIONS IN A SOFTLY BROKEN SUPERSYMMETRIC MODEL Kenzi TABATA, Isao UMEMURA and Katsuji YAMAMOTO 1 Department of Nuclear Engineering, Kyoto Uni~'ersity,Kyoto 606, Japan Received 7 June 1983
We discuss in an SU(3)C x SU(2)W X U(1)y gauge model with softly broken supersymmetry, the upper bounds on masses of the Higgs particle (mH) and the quark of the fourth generation (rnu) , and also the lower bound on Higgs mass. Requiring that no interaction should become strong until the grand unification energy is reached, we obtain the upper bounds m H < 220 GeV and m U < 150GeV. The lower bound on the Higgs mass is studied in detail for the case that the gaugino masses (Am) break supersymmetry softly. The vacuum is stable for any value of Am if mH>~ 12 GeV.
Recently, some interesting grand unified models have been investigated in detail on the basis of the supersymmetry which is broken spontaneously at the scale of Planck massMp [1] or at the intermediate scale (Mp .Mw)I/2 [2]. This is partly due to the fact that in these models with supersymmetry breakings at a rather large scale it is comparatively easy to reproduce relevant particle mass spectra at "low energies", and partly due to the expectation that these models might be derived from N = 1 supergravity theory [3]. A remarkable feature of these models is that the effects of the spontaneous breakdown of supersymmetry triggered in the "superheavy" fields appear as soft breaking terms for the "light" ( ~ O(Mw) ) fields at low energies. Hence it is likely that the effective low-energy theory has a softly broken supersymmetry. The purpose of this letter is to discuss some bounds on Higgs and fermion (quark and lepton) masses and on the softly breaking parameters of supersymmetry in the framework of the SU(3)C X SU(2)W × U(1)y gauge mode. From the viewpoint of perturbation theory, the upper bounds on Higgs and fermion masses are given by the requirement that the Higgs self-coupling and Yukawa couplings of fermions which obey the renormalization group equations should not increase too much until the i Address after July 1, 1983: Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina NC, USA. 80
grand unification scale is reached. In the supersymmetric model, the number of particles participating in interactions is much larger than that in the non-supersymmetric model and such a situation accelerates the increase of effective coupling constants with respect to the energy scale. Accordingly, the constraints on heavy quark and lepton masses in the supersymmetric model might be much more severe than those in the non-super symmetric model [4]. As for the lower bound on the mass of the Higgs particle, it can be derived from the requirement that the vacuum in which the SU(2)W X U(1)y gauge symmetry is spontaneously broken should be stable against radiative corrections [5]. Needless to say, the vacuum in the supersymmetric model never becomes unstable irrespective of the magnitude of the Higgs mass owing to the so-called "non-renormalization theorem". Therefore, the lower bound on the Higgs mass is given depending on the parameters of the supersymmetry breaking. Now, our model under consideration is constructed by chiral superfields, the transformation properties of which are shown with respect to SU(3)C X SU(3)w × U(1)y, they are shown in table 1, where the boldfaced (light-faced) number indicates the dimension (the quantum number) of the corresponding field for each group [U(1)y]. The superpotential of our model is as follows: W= Wn + Wy,
(la)
0.031-9163/83/0000 0000/$ 03.00 © 1983 North-Holland
15 September 1983
PHYSICS LETTERS
Volume 129, number 1,2
• dfE/dt=(fF/167r2)(h2
Table l SU(3) C
SU(2)W
U(1)y
1
2
1
1
2
1
tt N Q
1 3
1 2
0 1/3
Uc
3*
1
-4/3
Dc
3*
1
2/3
L Ec
1 1
2 1
+
3f~)2 + 4 f ~2 - 3 g w2- 3 g y )2.
(2g)
Here, the argument t of coupling constants is defined by t = 71 ln(q2/M 2 ) in terms of the momentum transfer squared q2 in each vertex, g c , g w and g y represent the SU(3)C , the SU(2)W and the U(1)y gauge coupling constants, respectively, and ng the number o f generations. We discuss the upper bounds on the masses o f the Higgs particle and the supposed quarks and leptons of the fourth generation. Since the Yukawa couplings of quarks and leptons of the three generations with the Higgs particle are small enough (mt ~ 35 GeV), their contributions to dh/dt, i.e., to the bounds on the mass o f the Higgs particle, can be ignored. When the Higgs field takes the following vacuum expectation value (VEV) _
Higgs
quark
lepton
H
- l 2
Wit = (h~IH - m2/2h) HN,
(lb)
(H O) = ( ~ 0 ) = o/X/~,
Wy = f u HQUC + fD HQDC + J ) [qLEc '
( 1c)
the masses of the Higgs particle, quarks and leptons are given by
where (h f ) and m 2 are dimensionless and dimensiontwo real coupling constants, respectively. Our lagrangian is composed of the terms~1 (gz*Taz) 2 + iOW/Ozi[2 preserving the supersymmetry and those breaking the supersymmetry softly, the latter of which is, as will be discussed later, characterized by the coupling constants with the dimension of mass. Therefore, the softly breaking terms do not take part in the renormalization group equations for the self-coupling constant h o f Higgs scalars and Yukawa coupling constants (/15, fi3 ,Jr,;) which determine the upper bounds on these coupling constants, in other words, the upper bounds on the masses of Higgs particle and fermions. Renormalization group equations for various coupling constants in our model are given in the one-loop approximation as follows:
dg(,tdt = (g3/167r2) ( 9 + 2 n g ) ,
(2a)
dg w ~dr -- (g3/16rr 2) ( - 5 +2ng),
(2b)
dgy/dt = (g3/16~ "2 ) (1 +-15~ ng),
(2c)
-2 2 -2~ 3g w 2 g y2) , ( 2 d ) dh/dt=(h/161r 2 ) ( 4 h 2 +3Ju+3fl~+/1
dfu/dt =(Ju/167rZ)(hZ+6f2+f2D -5~;C' 6" 2 - 3 g 2 _ ~ g 2 ) ,
(2e) dfD/dt=(JD/167r2)(h2+fl~+6)~+¢i216 , 2 7 25 g Y - g g c3 3gw
m H =~
ha,
(3)
toO, L = (fQ,L/X/2) a,
(4)
where the value o f a is estimated to be 174 GeV from the four-body weak coupling constant G. The upper bounds on the coupling constants, h and fQ, k, are obtained by the requirement that the effective coupling constants should not become strong in the "physical" region. As the initial values to solve the renormalization group equations (2), the following "experimental" values o f two parameters, the QCD parameter AMS [6] and the Weinberg angle 0 w [7], are adopted here: AMg = (160+ 80100~MeV, sin20w (Mw) = 0.215 + 0.015.
(5)
If one tries to unify the three interactions,gc,g w and g y , at some mass scale ~ using initial values consistent with these "experimental" values, a set of values. a c (Mw) = 0.12, sin20w (Mw) = 0.230 and # = M G = 2.10 X 1016 GeV, are obtained. By requiring that "fine structure constants", h2/47r, f2/47r etc., should not exceed unity until M G is reached [case (a)], the allowed regions in the m H - m U plane are shown inside tile solid lines in fig. 1 for two cases, m U =rn D ---mE and m U 5"- m D , m E . Allowed regions by a silnilar condition for the best " experimental" values, sin20w (Mw) = 0,215 and a c (Mw) = 0.1, and a typical energy scale
(20 81
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300iH(GeV)
PHYSICS LETTERS
once the vacuum with broken gauge symmetry is fixed as the absolute minimum state o f the tree-level potential, the vacuum state is still the absolute nrinimum o f the effective potential in an arbitrary order o f radiative corrections owing to the non-renormalization theorem. However, the stability o f the vacuum with broken gauge symmetry is questionable when supersymmetry is broken. In order to understand what types o f supersymmetry breakings bring about the instability of the vacuunr with broken gauge symmetry characterized by ~b= o, it is necessary to examine the effective potential V(¢) including the one-loop contribution
- - ,
20(
u~l*nD'rn E
10C
15 September 1983
,, ',
"Attowed"
V 1(~) [81:
0 ---
I
50
J
100
150
rnu(GeV)
Fig. 1. (a) Solid line: ~ c ( M w ) = 0.12, sin20w(Mw ) = 0.230 and MG = 2.10 × 1016 GeV. (b) Dotted line: ac(M W) = 0.10, sin20w(Mw) = 0.215 andMp = 1.22 X 1019 GeV.
v(,/,) : v 0 ( ¢ ) + v t @ ) ,
v0(~) = Vl(~b) =
Mp [case (b)] [although the unification of the three interactions can not be realized with these best values of sin20w (Mw) and O~C(Mw)], are plotted inside the dotted lines in fig. 1 also. Thus, the upper bounds on m H and mtt are obtained as follows: m H < 220 GeV,
m U < 150 GeV.
(6)
The above bound m U < 150 GeV will give a severe restriction on such models as proposed by Ellis et al. [ 1], in which the existence o f heavy quarks is indispensable for giving negative (mass) 2 to the SU(2)W doublets o f Higgs scalars owing to radiative corrections (which are triggered by the gaugino masses or heavy scalar boson masses) so that the spontaneous breakdown of SU(2)w X (1)y should be induced. Next. we discuss the lower bound on the mass of the Higgs particle. As was shown by Linde and Weinberg [5], using the standard model without heavy fermions, if the self-coupling of Higgs scalar is too small the vacuuln with the symmetry restored by radiative corrections is more stable than that with broken symmetry and the breakdown o f the gauge symmetry SU(2) w X U ( l ) y -+ U(1)e m does not occur. Of course, the field H N may develop a VEV when the Higgs mass term is added as a soft breaking of supersymmetry, The effect o f H N can, however, be ignored since we are discussing the lower bound on the mass of the Higgs particle (h ~ g o ,
gw, gY). In the supersymmetric model, as is well known, 82
~1 rn2¢ 2 + 1 /~2~b4
(~2 = m2/o2),
1 ~ rn4(q~) In [rn2(¢)/A2], 64rr 2 i
(7)
where the summation £i is taken over all helicity states with mass eigenvalue m i (q~) (with negative sign for fermions) and A is a suitable mass parameter. It should be remarked that the effective potential V(4~) has the possibility to take another local minimum at ~ = 0 in addition to the one with the broken gauge symmetry (~ = o). As is easily seen from eq. (7), the contributions o f fermions to the one-loop potential are opposite to those of bosons and decrease as q~ increases. If fermions have large gauge invariant masses owing to the soft breaking of supersymmetry other than those obtained by the Higgs mechanism (e.g., such mass terms as AmX~, for gaugino X), the fermion contributions rn I (4~ + Am) In [rn~ (¢ + A rn)/A 2 ] to tire one-loop potential hardly depend on q~ and behave as a meaningless constant because c~2/Am 2 ~ 1 [we are discussing the behaviour of V(~) for rather small ¢]. Thus, the t'ermion contributions decouple practically from the Higgs potential, and in effect V(o) rises relatively to V(0). On the other hand, if scalar bosons acquire a large mass, the situation is completely inverted. Accordingly, the inequality V(0) - V(o) > 0 (stability condition of the vacuum with the broken gauge symmetry) would be altered only when the contributions coming from the added mass term of fermions (gauginos) overcome those from bosons (squarks and sleptons etc). Let us discuss the potential behaviour when tire su-
Volume 129, number 1,2
PHYSICS LETTERS
persymmetry breaking mass term ,tESB of gauginos is added to our lagrangian: d2SB = Arn~.ax a + Arn'X0X 0 + h.c.,
(8)
where Xa and )~0 represent superpartners (two-component Weyl spinor) of SU(2)W and U(1)y gauge fields. respectively. In order to obtain the expression for V(O) V(o) concretely, it is necessary to determine the complete form of V(¢) by imposing the following renormalization condition at the point 4~= o of the minimum of V(¢),
ov(O)la¢l<~=o=o,
a2 v ( o ) / a O 2 Io = o : 2 m 2
>0.
[gauginos X± , ~3 of SU(2) w and gaugino X0 of the U(1)y]. The mass matrix for the charged fermions, X+ and H'+, is (~,+, H+) ( A m \u2
v(~)= v0(~)+ 1 ~
64n 2 i
{m~(~){lnlm~(~)/m~(o)] 2}2 (1o)
;2)(~-),
(12)
and that for the neutral fermions, X0, X3 , ~0 and H0, is
(xo x3, rio ~0) X
(9) T h e e f f e c t i v e p o t e n t i a l t h u s o b t a i n e d is expressed as follows:
15 September 1983
r
Am' 0 iu I
0 Am iu 2
iu l iu 2 0
--iu2
L. iul
iul] --iu2| 0 ,
X3 "~0 ,(13)
o,
0
"~0
where H and H_represent the superpartners of Higgs scalars, H and H, respectively, and u t -~gy(b/2 and u 2 = g w ~ / 2 [9]. From eq. (12), the mass squared (Dirac mass) of charged fermions are obtained as follows:
Accordingly, the difference V(0) - V(o) is given by
71 [4u22 + Am2 -+ A m ( A m 2 + 8u2)1/2] •
V(O)
As for the neutral fermion masses, the masses squared (Majorana mass) are given as the roots of the following fourth order algebraic equation of ~¢
+
+
V(o)=~h2o 4 1 ~ 647r2 i
{m4(0){ln[m2(0)/m2(o)] _-}}
2m~(O)m~(o)--5rni(o)}. 1 4
(14)
K{K3 -- K214(u 2 +u~) + Am 2 + Am'2l (11)
Since we want to obtain tire lower bound on the mass of the Higgs particle by using the stability condition V(O) - V(o) > 0, the contributions from one-loop corrections of tire Higgs scalars and their superpartners can be neglected in comparison with those from the gauge bosons and their superpartners. The effects of quarks (leptons) and squarks (sleptons) can also be ignored if the nmss difference between them is small compared with the added masses, Am and Am', of gauginos (the contributions of quarks (leplons) and squarks (sleptons) to the one-loop effective potential cancel out mutually when there is no mass difference between them). After all, it is sufficient tk~r our present problem to take account of the contributions from the gauge bosons and their superpartners only. The masses of particles which receive the contributions from the supersymmetry breaking term (8) at the tree-level are those of the gauge fermions
+ ~: [4(u~ +uT) +4u 2 " A m '2 + 4 u 2 " A m 2 + A m 2 A m '21
(15)
40t 7 - A m ' - u~ "Am) 2} = 0. m t I(Ge<') I i
1',) 'AHowed"
- " J
[
/"
/ 2_ L
//
/
/
/"
//
/ ] 0
(~
•
i
. . . . . . . . . 50
' 100
-
1- B0 Am(OeV)
I:ig. 2. Allowed region for the Higgs mass (mH) as a function of the added gauge fermion mass (Am). 83
Volume 129, number 1,2
PHYSICS LETTERS
The solutions o f the above e q u a t i o n (15) are obtained numerically by setting A m = A m ' for simplicity. Finally, putting the masses o f fermions obtained above together with those of the massive gauge bosons and their scalar superpartners (4u 2, 4u22 , 4(u 2 + u 2 ) ) i n t o the inequality [ V ( 0 ) - V(o)> 0] expressed by eq. (11), we can obtain the lower b o u n d on the mass o f the Higgs particle (m H = x/'-iho). The allowed region for m H is plotted in the rn H - A m plane in fig. 2. F r o m fig. 2, the stability c o n d i t i o n of the vacuum with the broken gauge s y m m e t r y is satisfied for any values o f added masses, A m and A m ' , o f gauginos i f m H > 11.5 GeV.
References [1] J. Ellis, L. I b ~ e z and G.G. Ross, Phys. Lett. 113B (1982) 283. [2] R. Barbieri, S. Ferrara and D.V. Nanopoulos, Z. Phys. C13 (1982) 276;
84
[3]
[4] [5]
[6] [7]
[8] [9]
15 September 1983
M. Dine and W. Fischler, Nucl. Phys. B204 (1982) 346; S. Dimopoulos and S. Raby, Los Alamos preprint LA-UR82-1282; S. Polchinsky and L. Susskind, Phys. Rev. D26 (1982) 3661. A.H. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49 (1982) 970; R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. 119B (1982) 343; J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 121B (1983) 123. N. Cabibbo, I. Maiani, G. Parisi and R. Petronzio, NucL Phys. B158 (1979) 295. A.D. Linde, JETP Lett. 23 (1976) 64; S. Weinberg, Phys. Rev. Lett. 33 (1976) 294. A.J. Buras, Rapporteur talk 198l Bonn Symp. on Lepton photon interactions at high energies. A. Sirlin and W. Marciano, Nucl. Phys. B189 (1981) 442; Ch. Llewellyn Smith and J.F. Wheater, Phys. Lett. 105B (1981) 486. S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; A. Weinberg, Phys. Rev. D7 (1973) 2887. J. Ellis and G.G. Ross, Phys. Lett. l17B (1982) 397.
PHYSICS LETTERS
15 September 1983
BOUNDS ON MASSES OF HIGGS BOSON AND HEAVY FEILMIONS IN A SOFTLY BROKEN SUPERSYMMETRIC MODEL Kenzi TABATA, Isao UMEMURA and Katsuji YAMAMOTO 1 Department of Nuclear Engineering, Kyoto Uni~'ersity,Kyoto 606, Japan Received 7 June 1983
We discuss in an SU(3)C x SU(2)W X U(1)y gauge model with softly broken supersymmetry, the upper bounds on masses of the Higgs particle (mH) and the quark of the fourth generation (rnu) , and also the lower bound on Higgs mass. Requiring that no interaction should become strong until the grand unification energy is reached, we obtain the upper bounds m H < 220 GeV and m U < 150GeV. The lower bound on the Higgs mass is studied in detail for the case that the gaugino masses (Am) break supersymmetry softly. The vacuum is stable for any value of Am if mH>~ 12 GeV.
Recently, some interesting grand unified models have been investigated in detail on the basis of the supersymmetry which is broken spontaneously at the scale of Planck massMp [1] or at the intermediate scale (Mp .Mw)I/2 [2]. This is partly due to the fact that in these models with supersymmetry breakings at a rather large scale it is comparatively easy to reproduce relevant particle mass spectra at "low energies", and partly due to the expectation that these models might be derived from N = 1 supergravity theory [3]. A remarkable feature of these models is that the effects of the spontaneous breakdown of supersymmetry triggered in the "superheavy" fields appear as soft breaking terms for the "light" ( ~ O(Mw) ) fields at low energies. Hence it is likely that the effective low-energy theory has a softly broken supersymmetry. The purpose of this letter is to discuss some bounds on Higgs and fermion (quark and lepton) masses and on the softly breaking parameters of supersymmetry in the framework of the SU(3)C X SU(2)W × U(1)y gauge mode. From the viewpoint of perturbation theory, the upper bounds on Higgs and fermion masses are given by the requirement that the Higgs self-coupling and Yukawa couplings of fermions which obey the renormalization group equations should not increase too much until the i Address after July 1, 1983: Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina NC, USA. 80
grand unification scale is reached. In the supersymmetric model, the number of particles participating in interactions is much larger than that in the non-supersymmetric model and such a situation accelerates the increase of effective coupling constants with respect to the energy scale. Accordingly, the constraints on heavy quark and lepton masses in the supersymmetric model might be much more severe than those in the non-super symmetric model [4]. As for the lower bound on the mass of the Higgs particle, it can be derived from the requirement that the vacuum in which the SU(2)W X U(1)y gauge symmetry is spontaneously broken should be stable against radiative corrections [5]. Needless to say, the vacuum in the supersymmetric model never becomes unstable irrespective of the magnitude of the Higgs mass owing to the so-called "non-renormalization theorem". Therefore, the lower bound on the Higgs mass is given depending on the parameters of the supersymmetry breaking. Now, our model under consideration is constructed by chiral superfields, the transformation properties of which are shown with respect to SU(3)C X SU(3)w × U(1)y, they are shown in table 1, where the boldfaced (light-faced) number indicates the dimension (the quantum number) of the corresponding field for each group [U(1)y]. The superpotential of our model is as follows: W= Wn + Wy,
(la)
0.031-9163/83/0000 0000/$ 03.00 © 1983 North-Holland
15 September 1983
PHYSICS LETTERS
Volume 129, number 1,2
• dfE/dt=(fF/167r2)(h2
Table l SU(3) C
SU(2)W
U(1)y
1
2
1
1
2
1
tt N Q
1 3
1 2
0 1/3
Uc
3*
1
-4/3
Dc
3*
1
2/3
L Ec
1 1
2 1
+
3f~)2 + 4 f ~2 - 3 g w2- 3 g y )2.
(2g)
Here, the argument t of coupling constants is defined by t = 71 ln(q2/M 2 ) in terms of the momentum transfer squared q2 in each vertex, g c , g w and g y represent the SU(3)C , the SU(2)W and the U(1)y gauge coupling constants, respectively, and ng the number o f generations. We discuss the upper bounds on the masses o f the Higgs particle and the supposed quarks and leptons of the fourth generation. Since the Yukawa couplings of quarks and leptons of the three generations with the Higgs particle are small enough (mt ~ 35 GeV), their contributions to dh/dt, i.e., to the bounds on the mass o f the Higgs particle, can be ignored. When the Higgs field takes the following vacuum expectation value (VEV) _
Higgs
quark
lepton
H
- l 2
Wit = (h~IH - m2/2h) HN,
(lb)
(H O) = ( ~ 0 ) = o/X/~,
Wy = f u HQUC + fD HQDC + J ) [qLEc '
( 1c)
the masses of the Higgs particle, quarks and leptons are given by
where (h f ) and m 2 are dimensionless and dimensiontwo real coupling constants, respectively. Our lagrangian is composed of the terms~1 (gz*Taz) 2 + iOW/Ozi[2 preserving the supersymmetry and those breaking the supersymmetry softly, the latter of which is, as will be discussed later, characterized by the coupling constants with the dimension of mass. Therefore, the softly breaking terms do not take part in the renormalization group equations for the self-coupling constant h o f Higgs scalars and Yukawa coupling constants (/15, fi3 ,Jr,;) which determine the upper bounds on these coupling constants, in other words, the upper bounds on the masses of Higgs particle and fermions. Renormalization group equations for various coupling constants in our model are given in the one-loop approximation as follows:
dg(,tdt = (g3/167r2) ( 9 + 2 n g ) ,
(2a)
dg w ~dr -- (g3/16rr 2) ( - 5 +2ng),
(2b)
dgy/dt = (g3/16~ "2 ) (1 +-15~ ng),
(2c)
-2 2 -2~ 3g w 2 g y2) , ( 2 d ) dh/dt=(h/161r 2 ) ( 4 h 2 +3Ju+3fl~+/1
dfu/dt =(Ju/167rZ)(hZ+6f2+f2D -5~;C' 6" 2 - 3 g 2 _ ~ g 2 ) ,
(2e) dfD/dt=(JD/167r2)(h2+fl~+6)~+¢i216 , 2 7 25 g Y - g g c3 3gw
m H =~
ha,
(3)
toO, L = (fQ,L/X/2) a,
(4)
where the value o f a is estimated to be 174 GeV from the four-body weak coupling constant G. The upper bounds on the coupling constants, h and fQ, k, are obtained by the requirement that the effective coupling constants should not become strong in the "physical" region. As the initial values to solve the renormalization group equations (2), the following "experimental" values o f two parameters, the QCD parameter AMS [6] and the Weinberg angle 0 w [7], are adopted here: AMg = (160+ 80100~MeV, sin20w (Mw) = 0.215 + 0.015.
(5)
If one tries to unify the three interactions,gc,g w and g y , at some mass scale ~ using initial values consistent with these "experimental" values, a set of values. a c (Mw) = 0.12, sin20w (Mw) = 0.230 and # = M G = 2.10 X 1016 GeV, are obtained. By requiring that "fine structure constants", h2/47r, f2/47r etc., should not exceed unity until M G is reached [case (a)], the allowed regions in the m H - m U plane are shown inside tile solid lines in fig. 1 for two cases, m U =rn D ---mE and m U 5"- m D , m E . Allowed regions by a silnilar condition for the best " experimental" values, sin20w (Mw) = 0,215 and a c (Mw) = 0.1, and a typical energy scale
(20 81
Volume 129, number 1,2
300iH(GeV)
PHYSICS LETTERS
once the vacuum with broken gauge symmetry is fixed as the absolute minimum state o f the tree-level potential, the vacuum state is still the absolute nrinimum o f the effective potential in an arbitrary order o f radiative corrections owing to the non-renormalization theorem. However, the stability o f the vacuum with broken gauge symmetry is questionable when supersymmetry is broken. In order to understand what types o f supersymmetry breakings bring about the instability of the vacuunr with broken gauge symmetry characterized by ~b= o, it is necessary to examine the effective potential V(¢) including the one-loop contribution
- - ,
20(
u~l*nD'rn E
10C
15 September 1983
,, ',
"Attowed"
V 1(~) [81:
0 ---
I
50
J
100
150
rnu(GeV)
Fig. 1. (a) Solid line: ~ c ( M w ) = 0.12, sin20w(Mw ) = 0.230 and MG = 2.10 × 1016 GeV. (b) Dotted line: ac(M W) = 0.10, sin20w(Mw) = 0.215 andMp = 1.22 X 1019 GeV.
v(,/,) : v 0 ( ¢ ) + v t @ ) ,
v0(~) = Vl(~b) =
Mp [case (b)] [although the unification of the three interactions can not be realized with these best values of sin20w (Mw) and O~C(Mw)], are plotted inside the dotted lines in fig. 1 also. Thus, the upper bounds on m H and mtt are obtained as follows: m H < 220 GeV,
m U < 150 GeV.
(6)
The above bound m U < 150 GeV will give a severe restriction on such models as proposed by Ellis et al. [ 1], in which the existence o f heavy quarks is indispensable for giving negative (mass) 2 to the SU(2)W doublets o f Higgs scalars owing to radiative corrections (which are triggered by the gaugino masses or heavy scalar boson masses) so that the spontaneous breakdown of SU(2)w X (1)y should be induced. Next. we discuss the lower bound on the mass of the Higgs particle. As was shown by Linde and Weinberg [5], using the standard model without heavy fermions, if the self-coupling of Higgs scalar is too small the vacuuln with the symmetry restored by radiative corrections is more stable than that with broken symmetry and the breakdown o f the gauge symmetry SU(2) w X U ( l ) y -+ U(1)e m does not occur. Of course, the field H N may develop a VEV when the Higgs mass term is added as a soft breaking of supersymmetry, The effect o f H N can, however, be ignored since we are discussing the lower bound on the mass of the Higgs particle (h ~ g o ,
gw, gY). In the supersymmetric model, as is well known, 82
~1 rn2¢ 2 + 1 /~2~b4
(~2 = m2/o2),
1 ~ rn4(q~) In [rn2(¢)/A2], 64rr 2 i
(7)
where the summation £i is taken over all helicity states with mass eigenvalue m i (q~) (with negative sign for fermions) and A is a suitable mass parameter. It should be remarked that the effective potential V(4~) has the possibility to take another local minimum at ~ = 0 in addition to the one with the broken gauge symmetry (~ = o). As is easily seen from eq. (7), the contributions o f fermions to the one-loop potential are opposite to those of bosons and decrease as q~ increases. If fermions have large gauge invariant masses owing to the soft breaking of supersymmetry other than those obtained by the Higgs mechanism (e.g., such mass terms as AmX~, for gaugino X), the fermion contributions rn I (4~ + Am) In [rn~ (¢ + A rn)/A 2 ] to tire one-loop potential hardly depend on q~ and behave as a meaningless constant because c~2/Am 2 ~ 1 [we are discussing the behaviour of V(~) for rather small ¢]. Thus, the t'ermion contributions decouple practically from the Higgs potential, and in effect V(o) rises relatively to V(0). On the other hand, if scalar bosons acquire a large mass, the situation is completely inverted. Accordingly, the inequality V(0) - V(o) > 0 (stability condition of the vacuum with the broken gauge symmetry) would be altered only when the contributions coming from the added mass term of fermions (gauginos) overcome those from bosons (squarks and sleptons etc). Let us discuss the potential behaviour when tire su-
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PHYSICS LETTERS
persymmetry breaking mass term ,tESB of gauginos is added to our lagrangian: d2SB = Arn~.ax a + Arn'X0X 0 + h.c.,
(8)
where Xa and )~0 represent superpartners (two-component Weyl spinor) of SU(2)W and U(1)y gauge fields. respectively. In order to obtain the expression for V(O) V(o) concretely, it is necessary to determine the complete form of V(¢) by imposing the following renormalization condition at the point 4~= o of the minimum of V(¢),
ov(O)la¢l<~=o=o,
a2 v ( o ) / a O 2 Io = o : 2 m 2
>0.
[gauginos X± , ~3 of SU(2) w and gaugino X0 of the U(1)y]. The mass matrix for the charged fermions, X+ and H'+, is (~,+, H+) ( A m \u2
v(~)= v0(~)+ 1 ~
64n 2 i
{m~(~){lnlm~(~)/m~(o)] 2}2 (1o)
;2)(~-),
(12)
and that for the neutral fermions, X0, X3 , ~0 and H0, is
(xo x3, rio ~0) X
(9) T h e e f f e c t i v e p o t e n t i a l t h u s o b t a i n e d is expressed as follows:
15 September 1983
r
Am' 0 iu I
0 Am iu 2
iu l iu 2 0
--iu2
L. iul
iul] --iu2| 0 ,
X3 "~0 ,(13)
o,
0
"~0
where H and H_represent the superpartners of Higgs scalars, H and H, respectively, and u t -~gy(b/2 and u 2 = g w ~ / 2 [9]. From eq. (12), the mass squared (Dirac mass) of charged fermions are obtained as follows:
Accordingly, the difference V(0) - V(o) is given by
71 [4u22 + Am2 -+ A m ( A m 2 + 8u2)1/2] •
V(O)
As for the neutral fermion masses, the masses squared (Majorana mass) are given as the roots of the following fourth order algebraic equation of ~¢
+
+
V(o)=~h2o 4 1 ~ 647r2 i
{m4(0){ln[m2(0)/m2(o)] _-}}
2m~(O)m~(o)--5rni(o)}. 1 4
(14)
K{K3 -- K214(u 2 +u~) + Am 2 + Am'2l (11)
Since we want to obtain tire lower bound on the mass of the Higgs particle by using the stability condition V(O) - V(o) > 0, the contributions from one-loop corrections of tire Higgs scalars and their superpartners can be neglected in comparison with those from the gauge bosons and their superpartners. The effects of quarks (leptons) and squarks (sleptons) can also be ignored if the nmss difference between them is small compared with the added masses, Am and Am', of gauginos (the contributions of quarks (leplons) and squarks (sleptons) to the one-loop effective potential cancel out mutually when there is no mass difference between them). After all, it is sufficient tk~r our present problem to take account of the contributions from the gauge bosons and their superpartners only. The masses of particles which receive the contributions from the supersymmetry breaking term (8) at the tree-level are those of the gauge fermions
+ ~: [4(u~ +uT) +4u 2 " A m '2 + 4 u 2 " A m 2 + A m 2 A m '21
(15)
40t 7 - A m ' - u~ "Am) 2} = 0. m t I(Ge<') I i
1',) 'AHowed"
- " J
[
/"
/ 2_ L
//
/
/
/"
//
/ ] 0
(~
•
i
. . . . . . . . . 50
' 100
-
1- B0 Am(OeV)
I:ig. 2. Allowed region for the Higgs mass (mH) as a function of the added gauge fermion mass (Am). 83
Volume 129, number 1,2
PHYSICS LETTERS
The solutions o f the above e q u a t i o n (15) are obtained numerically by setting A m = A m ' for simplicity. Finally, putting the masses o f fermions obtained above together with those of the massive gauge bosons and their scalar superpartners (4u 2, 4u22 , 4(u 2 + u 2 ) ) i n t o the inequality [ V ( 0 ) - V(o)> 0] expressed by eq. (11), we can obtain the lower b o u n d on the mass o f the Higgs particle (m H = x/'-iho). The allowed region for m H is plotted in the rn H - A m plane in fig. 2. F r o m fig. 2, the stability c o n d i t i o n of the vacuum with the broken gauge s y m m e t r y is satisfied for any values o f added masses, A m and A m ' , o f gauginos i f m H > 11.5 GeV.
References [1] J. Ellis, L. I b ~ e z and G.G. Ross, Phys. Lett. 113B (1982) 283. [2] R. Barbieri, S. Ferrara and D.V. Nanopoulos, Z. Phys. C13 (1982) 276;
84
[3]
[4] [5]
[6] [7]
[8] [9]
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M. Dine and W. Fischler, Nucl. Phys. B204 (1982) 346; S. Dimopoulos and S. Raby, Los Alamos preprint LA-UR82-1282; S. Polchinsky and L. Susskind, Phys. Rev. D26 (1982) 3661. A.H. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49 (1982) 970; R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. 119B (1982) 343; J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 121B (1983) 123. N. Cabibbo, I. Maiani, G. Parisi and R. Petronzio, NucL Phys. B158 (1979) 295. A.D. Linde, JETP Lett. 23 (1976) 64; S. Weinberg, Phys. Rev. Lett. 33 (1976) 294. A.J. Buras, Rapporteur talk 198l Bonn Symp. on Lepton photon interactions at high energies. A. Sirlin and W. Marciano, Nucl. Phys. B189 (1981) 442; Ch. Llewellyn Smith and J.F. Wheater, Phys. Lett. 105B (1981) 486. S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; A. Weinberg, Phys. Rev. D7 (1973) 2887. J. Ellis and G.G. Ross, Phys. Lett. l17B (1982) 397.