arXiv:hep-ph/9701370v2 14 Jun 1997. T M U P-H EL-9701. January,1997. R evised 4/4/97. A Supersym m etric C om posite M odel w ith D ynam icalSupersym ...
T M U P-H EL-9701 January,1997 R evised 4/4/97
A Supersym m etric C om posite M odel w ith D ynam icalSupersym m etry B reaking
arXiv:hep-ph/9701370v2 14 Jun 1997
N oriakiK itazawa and N obuchika O kada y z D epartm ent ofPhysics,Tokyo M etropolitan U niversity, H achioji-shi,Tokyo 192-03,Japan
A bstract W e present a supersym m etric com posite m odelw ith dynam icalsupersym m etry breaking. T he m odelis based on the gauge group SU (2)S SU (3)c
SU (2)L
SU (2)H
U (1)Y . Supersym m etry is dynam ically broken by the
non-perturbative e ect ofthe SU (2)S ‘supercolor’interaction. T he large top Yukawa coupling is naturally generated by the SU (2)H ‘hypercolor’ interaction as recently proposed by N elson and Strassler. T he supersym m etry breaking ism ediated to the standard m odelsector by a new m echanism . T he electroweak sym m etry breaking is caused by the radiative correction due to the large top Yukawa coupling w ith the supersym m etry breaking. T his is the ‘radiative breaking scenario’,w hich originates from the dynam ics ofthe supercolor and hypercolor gauge interactions.
e-m ail: kitazawa@ phys.m etro-u.ac.jp ye-m
ail: n-okada@ phys.m etro-u.ac.jp
zJSPS
R esearch Fellow
1
I. IN T R O D U C T IO N
In m ostofthe supersym m etric m odels,ifsupersym m etry isbroken,theelectroweak sym m etry breaking is caused by the radiative correction due to the large top Yukawa coupling. T his m echanism is know n as the ‘radiative breaking scenario’[1]. H owever,to understand the scenario com pletely, we should investigate two subjects. O ne is a m echanism of the supersym m etry breaking,and the other is a naturalexplanation ofthe large top Yukawa coupling. T he m ethod proposed by Seiberg etal.[2]is rem arkable,w hen we consider the dynam icalsupersym m etry breaking in (N = 1) supersym m etric gauge theories. By the m ethod,we can evaluate the non-perturbative e ect of a strong gauge interaction and exactly determ ine the dynam ically generated superpotential. T here are m any m odels [2-5]in w hich the supersym m etry is broken by the non-perturbative e ect ofthe strong gauge interaction. O n the other hand,the supersym m etric com posite m odelrecently proposed by N elson and Strassler[6]isrem arkableto understand thelargetop Yukawa coupling.T hey introduce the SU (2) gauge interaction w ith six doublet super elds, the ‘preon’ super elds. T here is SU (6) SU (5)
U (1)R global sym m etry. T he SU (5) subgroup is gauged, and identi ed w ith GS M ,w here G S M = SU (3)c
SU (2)L
U (1)Y is the gauge group ofthe standard
m odel. T hen,the ve doublet super elds transform as 5 ofthe SU (5),and the last one is singlet. Below the dynam icalscale ofthe SU (2)gauge interaction,the preon super eldsare con ned into the super elds of5 and 10 ofSU (5),w hich are identi ed w ith the up-type H iggs and the ten dim ensionalsuper elds in the conventionalsupersym m etric SU (5) G U T fram ework. T he top Yukawa coupling is dynam ically generated by the non-perturbative e ectofthe SU (2)gauge interaction,and itcan be understood asthe exchange ofthe preon super elds in m icroscopicalpoint ofview . T herefore,the top Yukawa coupling is naturally expected as O (1). In thispaper,we present a supersym m etric com posite m odelw ith dynam icalsupersym m etry breaking. T he m odelis based on the gauge group SU (2)S 2
SU (2)H
GS M . W e
assum e that dynam icalscales ofthese gauge interaction are ordered as w here
S
and
H
S
H
EE W ,
are the dynam icalscale ofthe SU (2)S and SU (2)H gauge interactions,
respectively, and E E W is the scale of the electroweak sym m etry breaking. It is useful to separate the m odelinto three building blocksby these energy scales,nam ely,the dynam ical supersym m etry breaking sector,the preon sectorproposed by N elson and Strassler,and the standard m odelsector. T hedynam icalsupersym m etry breaking sectorisbased on theSU (2)S ‘supercolor’gauge group w ith fourdoubletand eightsingletsuper elds. T here isSP (2) U (1)R globalsym m etry. T he SU (2) subgroup is gauged,w hich is identi ed w ith SU (2)H . T herefore,there are som e SU (2)H doublet super elds in this sector. T he supersym m etry is dynam ically broken at the scale
S
by the non-perturbative e ect of the supercolor interaction. In addition,
the U (1)R sym m etry is broken by the e ective K ahler potentialgenerated by the Yukawa coupling in the superpotential. Both the scalar and ferm ion elds in the SU (2)H doublet super eldsgettheirm assesdue to the breaking ofthe supersym m etry and U (1)R sym m etry. T he preon sector is based on the SU (2)H ‘hypercolor’ gauge group w ith six doublet preon super elds as proposed by N elson and Strassler. T he large top Yukawa coupling is naturally generated by the non-perturbative e ect of the hypercolor gauge interaction. Since we focuson the relation am ong the dynam icalsupersym m etry breaking,the large top Yukawa coupling and the electroweak sym m etry breaking,we concentrate only on the third generation. T herefore,our m odelis a ‘toy m odel’,but it is possible to include the last two generations as the elem entary particles. T he supersym m etry breaking is m ediated to the preon sector and the standard m odel sectorthrough the SU (2)H
GS M gauge interaction. T he hypercolorgaugino and the scalar
preons get the soft breaking m asses though radiative corrections by the SU (2)H doublet elds in the dynam icalsupersym m etry breaking sector. H ere,the doublet elds play a role ofthe ‘m essenger elds’[3]. Scalar elds com posed by scalar preons in the standard m odel get m asses,because ofthe m asses ofthe scalar preons. O nce the hypercolor gaugino and scalarpreonsbecom em assive,thestandard m odelgauginosgetm assesthrough theradiative 3
correction,since the preon super eldshave both chargesofthe SU (2)H and G S M .H ere,the preon and scalar preon elds,w hich are the SU (2)H doublet elds in the preon sector,also play a role ofthe ‘m essenger elds’. T his is a new m echanism to give the standard m odel gauginos the soft breaking m asses. A t low energy w here the preon super elds are con ned, the m echanism can be understood as the m ixing between the gauginos and the com posite ferm ions in adjoint representation. T he electroweak sym m etry is broken by the e ect ofthe large top Yukawa coupling as sam e as the radiative breaking scenario. It m ust be noticed that the breaking is triggered by the dynam ics of SU (2)S
SU (2)H gauge interaction, since both the supersym m etry
breaking and large Yukawa coupling owe their origin to the dynam ics. In section II,we introduce the preon sector w hich is recently proposed by N elson and Strassler. In section III,the dynam icalsupersym m etry breaking sector is constructed. In section IV , we discuss the m ediation of the supersym m etry breaking to low energy. In section V , the standard m odel sector is discussed w ith an em phasis on the spectrum of the m ediated soft breaking m asses. In section V I,we show that the electroweak sym m etry breaking really occursin ourm odelby including the e ectofthe large top Yukawa coupling. In section V II, the proton decay caused by the colored H iggs in our m odel is discussed, and we show thatthere is no contradiction w ith the experim ent. In section V III,we brie y discuss the R -axion problem in our m odeland give an exam ple ofthe solution. In section IX ,we sum m arize ourm odeland com m enton the extension ofthe m odelto include the rst and second generations.
II.T H E P R E O N SE C T O R
In this section we construct the preon sector w hich is based on the m odelrecently proposed by N elson and Strassler [6]. O nly the third generation is considered as m entioned above. W e introduce the SU (2)H hypercolor gauge group w ith the six doubletpreon superelds. T he m axim alnon-anom alous globalsym m etry ofthis sector is SU (6)
4
U (1)R ,and
the subgroup SU (5)
GS M is gauged,w here G S M = SU (3)c
SU (2)L
U (1)Y is nothing
butthe gaugegroup ofthestandard m odel.T he preon super eldsP and N transform under SU (2)H
SU (5) as follow s. SU (2)H
SU (5)
P
2
5
N
2
1
Itisconvenient to decom pose the preon super eld P asP = (d h),w here d and h transform as (3;1; 1=3)and (1;2;1=2),respectively,under the standard m odelgauge group GS M . In addition,we introduce two elem entary super elds 5 representation ofthe SU (5). T he eld
1
2
and
2,both
ofw hich are in
= (b ‘) represents the super elds ofthe anti-
bottom quark and the lepton doublet‘= ( H iggs super eld
1
),and
2
iscorresponds to the dow n-type
= (D H ) w ith D = (3 ;1;1=3)and H = (1;2; 1=2).
A t high energy w here the hypercolor interaction is weak,the superpotentialis generally given by (including dim ension 4 operators) W = +
H [hN ]+ b
M
D
H [dh]b+
b
D [dN ] ‘
M
H [hh]‘;
(1)
‘
w here square brackets denote the contraction ofthe SU (2)H index by the -tensor,and M b and M
‘
are param eters w ith dim ension one. A lthough the third and fourth term s in eq.(1)
are needed to generate the bottom and tau Yukawa couplings,we neglectthese term sin the follow ing,and consider only the top Yukawa coupling. Below thescale
H
w heretheSU (2)H interaction becom esstrong,thistheory isdescribed
by the m assless SU (2)H singlet e ective super elds M
ij,w here
i and j are avor indices.
T he e ective super eld is the 15-com ponent antisym m etric tensor given by
M =
0
1
0
1
B B B B B B @
dC
B B B B B B @
[dd] [dh] [dN ]C
h
C C C C C A
=
d h N
[hd] [hh] [hN [N d] [N h] 0
N 5
C C ]CC C A
;
(2)
w here
and
are the SU (2)H indices. T his tensor is decom posed as 5 + 10 ofSU (5). 0
1
B
[dN ]C
5 :B@
C A
0
1
B
[dd] [dh]C
; 10 :B@
C A
(3)
:
[hd] [hh]
[hN ]
H ere,the super eld in 5 representation can be identi ed w ith the up-type H iggs super eld ([dN ][hN ])
H
(D H ),w here D = (3;1; 1=3) and H = (1;2;1=2). T he com ponents of
the super eld in 10 representation are also identi ed as [dd] and [hh]
H
H
t,[dh]
H
(tb)=
H
q,
,respectively.
T he dynam ically generated superpotentialis exactly given by W
dyn
1
=
3 H
Pf(M )=
1
1
3 H
233!
ijklm n
M ijM klM
mn
:
(4)
>From eqs.(1) and (4),the totale ective superpotentialis given by W
N ote that
eff
=
H qt+
qqD +
+
HH +
D
tD
DD :
(5)
O (1) is naturally expected,since it can be understood that these
Yukawa couplings are constructed by the exchange ofthe preon super elds. T he rst term in eq.(5) is nothing but the top Yukawa coupling,w hich is naturally large. T he fourth and fth term s are the -term s ofthe SU (2)L doublet and the color triplet H iggs super elds, respectively,w here
H
,and
D
H
D
from eq.(1).
N ote thatthiscom posite m odelisthe only one w hich satisfy the follow ing three requirem ents. T he rst is that the m odelis based on the gauge group SU (N ) w ith vector-like N f m atter elds in the fundam entalrepresentation. T he second is that there is only the ‘m esonic’ e ective super elds but not the ‘baryonic’ one. T he third is that the Yukawa coupling has to be dynam ically generated. T hese can be satis ed only ifN f = N + 1 and N = 2.
6
III.T H E D Y N A M IC A L SU P E R SY M M E T R Y B R E A K IN G SE C T O R
In orderforthe supersym m etric com posite m odeldiscussed in the previoussection to be realistic,the supersym m etry should be broken. W e constructthe dynam icalsupersym m etry breaking sector in this section. Itisrecently pointed outthatthetheory oftheSU (2)gaugeinteraction w ith fourdoublet and six singletsuper eldsbreakssupersym m etry by the non-perturbative e ectofthe gauge interaction [5]. T he dynam icalsupersym m etry breaking sector in our m odelhas the sam e structure as the theory. T his sector is based on the SU (2)S supercolor gauge group w ith four doublet and eight singlet super elds. T he m axim alnon-anom alous globalsym m etry ofthis sector is SP (2) U (1)R . T he SU (2) subgroup ofSP (2) is gauged,and identi ed w ith the hypercolor gauge group SU (2)H . N ote that we treat the hypercolor interaction as weak in the discussion of the dynam icalsym m etry breaking,because ofthe assum ption
S
H
.
T he particle contents are as follow s. SU (2)S
SU (2)H
U (1)R
Q
2
2
0
Q~1
2
1
0
Q~2
2
1
0
Z
1
1
2
1
1
2
Z1
1
2
2
Z2
1
2
2
X
1
1
0
Y
1
1
2
Z
0
T he tree levelsuperpotentialis introduced as W
tree
=
tr[Z^V^ ]+ 4
V
ZV +
7
ZZX
2
+
Y
YX 2 ;
(6)
w here Z^ and V^ are the antisym m etric 4
4 m atrices de ned by 3
2
Z^ =
0
6 6 6 6 6 6 6 6 6 6 4
Z+Z 0
(Z + Z )
0
Z12
Z21
Z22
Z 21
Z 12
Z 22
Z1
0
Z11
1
0 (Z
0
Z 0
Z)
Z
7 7 7 7 7 7 7 7 7 7 5
(7)
0
and
V^ =
0
1
B B B B B B @
Q C C C
Q~1 CC Q~2
Q Q~1 Q~2
C A
2
=
3
0
6 6 6 6 6 6 6 6 6 6 4
V +V 0
(V + V )
0
0
V11
V12
V21
V22
V11
V21
V12
V22
0 (V
0
V 0
V)
V
7 7 7 7 7 7 7 7 7 7 5
;
(8)
0
respectively. H ere, a = 1;2 of Z ia and Via (i = 1;2) is the SU (2)H index, and
and
are the SU (2)S indices. T he eld Z^ is in the reducible representation 1 + 5 ofSP (2): Z is 0 singlet,and Z and Z i are in 5. T his is the sam e for the eld V^ . N ote that eq.(6) is the
generalsuperpotentialw hich possesses the SP (2) Below the scale
S
U (1)R globalsym m etry.
w here the supercolor interaction becom es strong,this sector is de-
scribed by the e ective super eld V^ . T he dynam ically generated superpotentialis exactly given by W
dyn
= A (PfV^ = A (V 2
4 S
)
0
V2
[V1 V2]
4 S
);
(9)
w hereA istheLagrangem ultipliersuper eld,and thesquarebracketsdenotethecontraction of the SU (2)H indices w ith the -tensor. T he total e ective superpotential is given by W
eff
= W
tree +
W
dyn .
8
In the analysisofthe vacuum ,eq.(9)isregarded asa constraint,PfV^ constraint one super eld should be elim inated from W super eld V from W
eff
eff . W
4 S
= 0.By this
e elim inate the SP (2) singlet
by the constraint,since it is expected that the vacuum is realized
w ith the largest globalsym m etry. T hen,the e ective superpotentialis rew ritten by q
W
=
eff
4(
V
n
0
4
)Z
S
+ V 02 + [V1 V2] o
0
4Z V + 2[Z 1 V1]+ 2[Z 2 V2] +
ZZX
2
+
Y
YX 2 ;
(10)
w here the am biguity ofthe sign in front ofthe square root is absorbed by the U (1)R phase rotation. By factoring out
S
to norm alize the dim ension ofthe e ective super elds,and
expanding the square root,we obtain W
eff
’
4(
V
n S
)Z 0
2 S
+
1 02 1 V + [V1 V2] 2 2 o
0
4Z V + 2[Z 1 V1]+ 2[Z 2 V2] +
ZZX
2
+
Y
YX 2 :
(11)
T his e ective superpotentialis one ofthe type ofO ’R aifeartaigh m odels[7],and ve supersym m etric vacuum conditions,@W and @W
eff =@Y
eff =@Z
= 0,@W
eff =@Z
0
= 0,@W
eff =@Z i
= 0 (i= 1;2)
= 0,cannot be satis ed sim ultaneously. A s a result,the supersym m etry is
broken. N ote thatthe e ective potentialhasthe ‘pseudo- atdirection’asO ’R aifeartaigh m odels have. For exam ple,the vacuum energy rem ains m inim um along the direction ofarbitrary vacuum expectation value hZ i 1 ,w hen the other elds have no vacuum expectation values. H owever,this atnessw illdisappear,ifthe quantum correction to the K ahlerpotential is considered. T he e ective K ahler potentialat low energy cannot be reliably evaluated, because ofthe strong supercolor interaction. T herefore,at the high energy w here the supercolor is weak and negligible,we evaluate it by considering only the Yukawa couplings in the tree levelsuperpotential. N ote that the super eld Z couples to the m assless super eld X in eq.(6).
1
W e use the sam e notation for the super eld itselfand the scalar com ponent ofthe super eld.
9
In the non-supersym m etric m assless
4
theory,itiswellknow n thatthe infrared diver-
gence appears in the loop correction ofthe m assless eld. A lthough the infrared divergence disappears w ith sum m ing up the 1-loop diagram s,the potentialw ith the 1-loop correction becom es singular at the point h i= 0,as a result. T he problem ofthe infrared divergence or the singularity at the origin is solved by the fact that the non-zero vacuum expectation value h i6 = 0 em ergesin the e ective potential. T hism echanism ispointed outby C olem an and W einberg [8]. W hen the e ective K ahler potentialofthe super eld Z is evaluated by considering the loop correction ofthe m assless super eld X ,we encounter the sam e situation w ith the infrared divergence,since the super eld istreated asthe boson eld in the supergraph m ethod [9].U nfortunately,the 1-loop e ective K ahlerpotentialisnotreliable (see A ppendix). H ow ever,we can expectthatthe e ective K ahlerpotentialofZ isnon-trivial,and the m inim um ofthe e ective potentialis realized at the point hZ i6 = 0
2
,according to the m echanism of
C olem an and W einberg. T his discussion is also applied to the e ective K ahler potentialof the super eld Y ,and hY i6 = 0 is also expected. In thefollow ing,we assum e thatK ahlerpotentialsofallthe eldsexceptforZ and Y are naive,and theirscalarcom ponentshave no vacuum expectation values. By thisassum ption, the vacuum energy is given by , "
E vac = 16(
2
V
)
4 S
!#
@K Z @ (Z yZ ) y @(Z Z ) @(Z yZ )
;
(12)
Z = hZ i
w here K Z is the non-trivialK ahler potentialof the super eld Z . T he supersym m etry is broken by the SU (2)S supercolorgaugedynam ics,unlessthe denom inatorbecom essingular.
2
T he possibility that the scalar com ponent ofZ has a large vacuum expectation value w ithout
the additionalYukawa coupling
Z
ispointed outby Shirm an [10]. Ifitistrue,we can starta m ore
sim ple SP (2) U (1)R sym m etric superpotential,W the super elds X and Y .
10
tree
=
tr[Z^V^ ]+ 4
V
Z V ,w ithoutintroducing
IV . T H E M E D IA T IO N O F SU P E R SY M M E T R Y B R E A K IN G
In thissection,we investigate the m ediation ofthe supersym m etry breaking. T he breaking ism ediated to thepreon sectorand thestandard m odelsectorby theradiativecorrections due to the SU (2)H and G S M charged particles. T he SU (2)H doublet elds play the role of the ‘m essenger elds’[3]. Letusconsiderthe m assspectrum ofthe SU (2)H doublet elds from the superpotential ofeq.(11). T hese are given by 0
(m2 + m 2)(V1yV1 + V2yV2)
L m ass =
0
m 2(Z 1yZ 1 + Z 2yZ 2) m [ V1
m ([
(13)
0
m m (Z 1yV2 + Z 2yV1)+ h:c:
0
V2 ]
F [V1 V2]+ h:c:
V1 ]+
Z1
[
Z2
V2 ])+
h:c:;
w here !#
, "
F m 0
m
2
8(
V
2( 2
V
S
@K Z @ (Z yZ ) y @(Z Z ) @(Z yZ )
2 S
)
(14)
; Z = hZ i
)hZ i;
:
W e can regard F , m and m
0
as three independent free param eters instead of the four
independent free param eters ,
V
,
Z
and
S
in the follow ing 3 . N ote that the SU (2)H
doublet super elds Z i and Vi (i= 1;2) should be decoupled at the energy scale of
H
,in
orderforthe m echanism by N elson and Strasslerreally to works. T hisfactdem andsto take 2 H
0
F;m 2,and m 2. In the follow ing discussion,we set m
0
m and F
m 2,and treat
the SU (2)H coupling perturbativly. T he hypercolorgaugino getsitssoftbreaking m assthrough the 1-loop diagram in Fig.1. It is given by
3
N ote that the K ahler potentialK Z and the vacuum expectation value hZ i are determ ined by
the coupling
Z.
11
m w here
H
F ; 8 m H
H
(15)
= gH2 =4 is the coupling constant ofthe hypercolor interaction. T his is the rst
order correction of
H
.
T he scalarpreons,P~ and N~,also gettheirm assesthrough the 2-loop diagram sin Fig.2. Since their SU (2)H charge is the sam e,they have the sam e m asses given by 3 H 2 4
m 2P~ = m 2N~ T his is the second order correction of
H
2
2
F m
(16)
:
. N ote that the scalar preon m asses are the sam e
orderofthehypercolorgaugino m ass.Sincethescalarpreonshavethem asses,thecom posite scalars, the scalar quarks q ~ and ~ t, the scalar lepton ~ , and the up-type H iggs boson also have the sam e soft breaking m asses. W e assum e that their m asses ofthe com posite scalars are tw ice ofthe scalar preon m ass,for sim plicity. Furtherm ore,the gauginosin the standard m odelgettheirm asses through the radiative correction including the preon super eld P ,since it has not only SU (2)H charge but also G S M charge. By 3-loop diagram m w here
N
N
4
in Fig.3,the m ass is given by 3 N 8
H
4
2
F ln m
m
(17)
;
H
(N = 1;2;3) is the coupling constant ofthe SU (N ) interaction in the standard
m odel5 . T his 3-loop correction is the second order for
H
,and the rst for
N
.
N ote that the preon and scalar preon elds play the role of‘m essenger elds’,and this is a new m echanism to give soft breaking m asses to the standard m odelgauginos. A t low energy w here the preon and scalarpreon are con ned,the m echanism can be understood as
4
A sim ilar diagram was considered by L.R andallin ref.[11].
5
In thiscalculation,theinfrared divergence occurs,w hich m eansthatthem assatzero m om entum
cannot be de ned. W e have taken
H
as the cut o param eter ofthe infrared divergence,since
the preon super eld P are con ned at the energy lower than
12
H
.
the m ixing between gauginos and the com posite ferm ions in adjoint representation. T his is a specialpicture ofthis supersym m etric com posite m odel. W e have show n that the soft breaking m asses of the particles in the standard m odel are really generated. H owever,in the discussion ofthe electroweak sym m etry breaking,the above perturbative estim ation cannot be reliable,since the hypercolor interaction becom es strong at low energy. In the follow ing,we use the m asses given in eqs.(16) and (17) w ith H
=4 = 1 for order estim ation. T he condition m
H
;m P~
H
m ust be satis ed in order
for the m echanism by N elson and Strassler really to work at low energy.
V . T H E ST A N D A R D M O D E L SE C T O R
Let us consider the standard m odelsector at the scale w here the preon super elds are con ned by theSU (2)H interaction.T helargetop Yukawa coupling isgenerated asdiscussed in section II.T he scalarquarks(~ q and ~ t),the scalarlepton (~ )and the up-type H iggsboson have the sam e soft breaking m asses p m q~ = m ~t = m ~ = m H w here we use eq.(16) w ith by eq.(17)w ith
H
H
6
2mP~
F m
(18)
;
=4 = 1. T he standard m odelgauginos have m asses as given
=4 = 1.N otethattheratio ofthem assesoftwo gauginosaredeterm ined
by the ratio oftwo gauge coupling; m
N
=m
M
=
N
=
M
(19)
;
w here N ,M = 1;2;3. T his relation is the sam e as obtained in the gauge-m ediate m odelof ref.[3]. Furtherm ore,other soft breaking m asses in the standard m odelare also induced by the radiative correction through the standard m odelgauge interaction. By the 1-loop diagram in Fig.4,the m ass ofthe dow n-type H iggs boson H is given by "
m
2 H
1
8
2 D
ln 1 +
m 2q~ 2 D
!
+m
2 q ~
ln 13
2 D 2
+ m 2q~ + m 2q~
! 2
ln 1 +
m 2q~ 2
!#
:
(20)
T he soft breaking term B H H + h:c:in the H iggs potentialis also induced by the 1-loop diagram in Fig.5. W e obtain 3 2 m 2
B
! 2
ln
H
m
(21)
; 2
w here the ultraviolet divergence is cut o by the com posite scale 2
=m 22
H
, and we assum e
1.
By considering allthe estim ated soft breaking m asses,the H iggs potentialis given by V =
1 g2 + g22 2 1 2 2 ( + m H2 )h + ( 2 + m 2q~)h2 + B hh + 1 (h 2 2 32
h2)2 ;
(22)
w here g1 and g2 are the gauge coupling constants, and h and h are the real part of the neutralcom ponent ofH and H ,respectively. Since m 2H > 0 and B < m 2q~ from eqs.(17), 2
(18), (20), and (21) w ith
From eqs.(17),(18) and (21),we obtain , " p # 3 6 22 2 2 2 2 tan ( + m H )=B ( + mH) m q~ ln(r) ; 32 2 w here the term 1=2M
2 Z
cos2 isneglected w ith
=
(27)
200 G eV ,and the logarithm in eq.(21)
isconsidered asO (1).Substituting eq.(20)to (27),one can check the param eterdependence oftan
m entioned above. O n the other hand,we obtain from eq.(25) "
H
m q~ w here the relation m 2q~
2 2v2 tan2 exp 3m 2t 1 + tan2
#
(28)
;
2
, M Z2 , and B is used. T he value of
H
=m q~ has the sam e
dependence as tan ,since it is the increasing function oftan . By using m q~,the m ass ofthe gauginos in the standard m odelsector are given by p 6 N m N m q~ ln(r): (29) 16 T hen,we obtain the values ofthe m asses ofthe gauginos for m q~ = 20 TeV as m
3
810G eV ; m
2
220G eV ; m
1
110G eV :
(30)
T hese values are experim entally acceptable [12].
V II.T H E P R O T O N D E C A Y P R O B LE M
In our m odelthere are colored H iggs super elds D and D w hich m ay cause the proton decay. T he baryon num ber violating interaction at the tree level in the rst and second generations is forbidden by im posing usualU (1)B
L
singlets. O n the other hand,there is another U (1)B 16
sym m etry,under w hich D and D are L
sym m etry in the third generation,
under w hich D and D have charge 2=3 and
2=3, respectively. T his sym m etry allow s
the baryon num ber violating interactions as the second and third term s in eq.(5). T hese interactions m ay cause too rapid proton decay through the avor m ixing between the third and othergenerationsw hich reducesthese two sym m etriesto the unique U (1)B
L
sym m etry
6
. To discuss the problem of proton decay, we should know the quark and lepton m ass
m atrices,in other words,we need a m odelw hich explains the origin ofm asses ofallquarks and leptons. A lthough such a m odelis not yet constructed,we can discuss proton decay w ith som e assum ptions w hich can be reasonably expected in our m odel. In order to generate the up quark m ass and the m ixing between the up and top quarks, the follow ing non-renorm alizable interactions m ust be introduced in the superpotential. W =
uu
M
ut [hN 2 M ut
[hN ]q1u +
uu
]q1[dd]+ 2
H
M
H q1u +
uu
H
M
H q1t+
ut
tu [hN M t2u 2 H
M
to assum e that M
uu
M ut
uu
ut
(31)
H q3u ;
tu
w here qi denotesthe quark doubletin the i-th generation,M w ith dim ension one,and we take
][dh]u
uu ,M ut and
M
tu
are param eters
1 in the second line. It would be able
tu
M tu ,and the m ixing angle between the up and top quarks is
given by
ut
mu mt
2
:
(32)
T he sam e argum ent is applied to the second generation, and the m ixing angle between the charm and top quarks is given by (m c=m t)2. N ote that the avor m ixings are highly suppressed in the up-type quark sector.
6
Itispossible to identify
in eq.(5)w ith a singletheavy particle instead ofthe right-handed tau,
as done in the originalpaper by N elson and Strassler [6]. In this case,there in no problem on the proton decay.
17
O n theotherhand,forthe dow n-type quarksand charged leptons,we cannotobtain such relations,since H is the elem entary particle and no higher dim ensionalterm s are needed to introduce the Yukawa interactions. H owever,we can assum e thatthe m ixing angle between the dow n and bottom quarks is ofthe order ofthe (1,3) or (3,1) elem ent ofthe K obayashiKM M askawa m atrix ( 13
103 [12]),since the m ixing angles between the up-type quarks
are very sm allas assum ed above. Letusdiscuss the proton life tim e w hich iscaused by the four-ferm ion interaction m ediated by the colored H iggs and H iggsino. T he life tim e is roughly described as p
G D2 ; m 5p
(33)
w here G D is the coupling constant of the four-ferm ion interaction, and m p is the proton m ass. T he experim entalbound is obtained by 1032 G eV
GD < 2 from
p
0
( +) (
0 +
e ( + ) through lepton m ixing and
m eanso -shell),caused by the four-ferm ion interaction through the colored 0 +
e ( + ),G D is roughly estim ated as
H iggs exchange depicted in Fig.6. Forthe m ode p ! GD w here
(34)
> 1032 yr [12].
A t the tree level, there are decay m odes, p ! p!
2
e ( )
2 m 2D
2 KM ut 13
e ( )
(35)
;
isthem ixing anglebetween the right-handed electron (m uon)and tau,and m D
isthe colored H iggsm ass(note thatm 2D = m 2q~ +
2 D
from eqs.(5)and (18)).By substituting
the num ericalvalues used in section V Iand m u = 5 M eV ,we obtain G D T herefore,there is no contradiction w ith the experim ents,if p!
0
( + ) m ode,
e ( )
e ( )
is replaced by the factor ofthe o -shell
10 5 . W e obtain the su ciently sm allvalue,G
D
3
3
1030
e ( ).
< 10 2 . In the case of \decay",G F m 3p=m
1035 .
A t the 1-loop level, the dom inant decay m odes are p !
0 +
e ( + ) and p !
0
( +),
w hich are caused by the four-ferm ion interaction through the colored H iggsino exchange depicted in Fig.7. For the m ode p !
0 +
e ( + ),we obtain 18
3
GD
6 H ere,note that the dependence on
D
1 m 2q~
2 KM ut 13
1032
and m
e ( )
3
e ( )
(36)
:
is absent because of
2 D
;m 23
value of G D is allowed, if
e ( )
< 10 2 . T he m ode p !
suppressed,since the angle
e ( )
is replaced by the factor ofthe o -shell
m 2q~. T his
0
( + ) is also substantially \decay".
>From the follow ing argum ent,one can expect that the proton decay is su ciently suppressed in any process.Sincethebaryon num berviolating interactionsexistonly in thethird generation,the baryon num berviolating interactionsin the rstand second generationscan be induced only through the avor m ixing between the third and othergenerations. T herefore,in any proton decay process,G D isalwayssuppressed by the avorm ixing angles.T he angles in the up-type quark sector can be extrem ely sm allas reasonably expected above. M oreover, the angles for the avor m ixing of the scalar particles are very sm all, because ofthe decoupling e ect due to the heavy scalars in the third generation. T he value ofG D becom es very sm allby these sm allm ixing angles.
V III.T H E R -A X IO N P R O B LE M
In section III, we assum e that hZ i = 6 0 and hY i = 6 0. T his m eans the spontaneous breaking of the global U (1)R sym m etry, and the N am bu-G oldstone boson called the R axion appears. Since the quark super elds have non-zero R -charge, they couple to the R -axion. T his coupling m ay cause an astrophysicalor a cosm ologicalproblem ,such as an overcooling ofthe red giantsora breakdow n ofthe successfulpredictionsfrom the big bang nucleosynthesis. W e consider the possibility to avoid the problem . Let us introduce
X
X 3 term into the tree levelsuperpotentialofeq.(6). N ote that the
new term does not disturb the supersym m etry breaking. T he R -charge ofthe super elds are changed as follow s.
19
U (1)R V^
4/3
Z^
2/3
X
2/3
Y
2/3
N ote that the super elds Q and Q~i m ust have R -charge,because ofthe non-zero R -charge of V^ . T herefore, the new U (1)R sym m etry is explicitly broken by the SU (2)S supercolor anom aly,and the light R -axion disappears. H owever,the generality ofthe superpotentialis lost by this strategy. For exam ple,we cannot introduce
0
Y 3 term ,w hich is allowed by the
sym m etry,into the superpotential,since the term recovers the supersym m etry.
IX .SU M M A R Y A N D C O M M E N T S
W e have constructed a supersym m etric com posite m odelw ith dynam icalsupersym m etry breaking. T he m odelisbased on the gauge group SU (2)S
SU (2)H
GS M .T here are three
building blocks, nam ely, the dynam ical supersym m etry breaking sector, the preon sector recently proposed by N elson and Strassler,and the standard m odelsector. In thedynam icalsupersym m etry breaking sector,thenon-zero vacuum expectation value ofthe F com ponent ofthe gauge-singlet super eld Z arises by the non-perturbative e ect ofthe SU (2)S supercolor interaction. T herefore,supersym m etry is dynam ically broken. In addition,the scalarcom ponent ofZ isassum ed to have non-zero vacuum expectation value by the e ective K ahler potential including the radiative corrections due to the m assless gauge-singlet super eld X . T he U (1)R sym m etry is spontaneously broken by this vacuum expectation value. In thepreon sector,thepreon super eldsarecon ned into theup-typeH iggsin 5 and the super eld in 10 ofSU (5)
GS M by the non-perturbative e ect ofthe SU (2)H hypercolor
interaction. T he large top Yukawa coupling is dynam ically generated,and it is understood as the exchange ofthe preons in the top quark and H iggs doublet. 20
T he supersym m etry breaking is m ediated to the preon sector and the standard m odel sector by the radiative corrections. T he hypercolor gaugino gets the soft breaking m ass through the 1-loop diagram in Fig.1,and the scalar preons also get the m asses from the 2loop diagram sin Fig.2.T he softbreaking m assesofthe com posite scalars,the scalarquarks and lepton and the up-type H iggs boson,originate from the m ass ofthe scalar preons. T he gauginos in the standard m odelget the soft breaking m asses through the 3-loop diagram in Fig.3. It is crucialthat the preon super eld P has both charges ofthe hypercolor and the standard m odel gauge groups, and it plays a role of the ‘m essenger eld’ . T his is a new m echanism to generate the soft breaking m asses of the standard m odel gauginos. Below the scale w here the hypercolor interaction becom es strong,this m echanism can be understood as the m ixing between the gauginos and the m assive com posite ferm ions in adjoint representation. T his m odel predicts the spectrum of the soft breaking m asses in the standard m odel sector. A llcom posite scalar elds have the sam e soft m ass,and the m asses ofthe gauginos satisfy the relation of eq.(19). A ll possible soft breaking m asses in the H iggs potential are generated by the radiative correction due to the standard m odel gauge interactions. A lthough the H iggspotentialw ith allthe softbreaking m assescannotcause the electroweak sym m etry breaking,the breaking is realized by the e ect ofthe large top Yukawa coupling. T hisisthe radiative breaking scenario,w hich originatesfrom the dynam icsofthe supercolor and the hypercolor interactions. W e have show n that the electroweak sym m etry breaking can occurw ith experim entally acceptable valuesofthe softbreaking m assesin the standard m odel. In ourm odel,thereisthecolored H iggsw hich isstrongly coupled to quarksand lepton as eq.(5).Itseem sthattheinteraction causestoo rapid proton decay through the avorm ixing. H owever,in according to eq.(31),itcan be reasonably assum ed thatthe m ixing between the third and other generations is highly suppressed. A s a result,there is no dangerous proton decay. T hereisa potentialR -axion problem in them odel,sincethenon-anom alousU (1)R global 21
sym m etry is spontaneously broken by the non-zero vacuum expectation value ofthe scalar com ponent ofZ . For exam ple,this problem can be avoided by introducing the new term X
X 3 into the superpotential of eq.(6), because this term requires to change the charge
assignm ent ofU (1)R . Since the new U (1)R sym m etry is explicitly broken by the supercolor anom aly,no light R -axion em erges. H owever,the generality ofthe superpotentialis lost. Finally,we com m enton the avorchanging neutralcurrentbetween the rstand second generations. T he scalar partners in these generations can get the soft breaking m asses by radiative correctionsasin Fig.4 and/orFig.5.Since the scalarpartnersw ith the sam e gauge charge are degenerate,the avor changing neutralcurrent between these two generations is suppressed. T his feature is the sam e as the gauge-m ediate m odel[3].
A P P E N D IX A : T H E E F F E C T IV E K A H LE R P O T E N T IA L
W e discuss the e ective K ahler potentialofthe super eld Z generated by the Yukawa coupling
Z
in the superpotentialof eq.(6). W e calculate the in nite series of the 1-loop
graphs,and obtain the e ective K ahler potentialas "
K eff = Z yZ 1 w here
2 Z 8 2
ln
2 y ZZ Z 2
!#
(A 1)
;
is the scale introduced by the wave function renorm alization ofthe super eld Z .
T hen,the vacuum energy is given by @W E vac = @Z
2,
"
1
2 Z 8 2
(
2 + ln
2 y ZZ Z 2
!)#
:
(A 2)
Z = hZ i
O ne m ay expect that the vacuum is realized at hZ i = 0, since the denom inator diverges and E vac vanishes. H owever, the 1-loop approxim ation cannot be valid near the origin, since the logarithm becom es too large in eq.(A 1). A lthough the com plete calculation seem s to be im possible like in the non-supersym m etric m assless
4
theory discussed by C ole-
m an and W einberg, we expect that hZ i = 6 0 is realized. T he appearance of the non-zero vacuum expectation value seem s to be a com m on m echanism to solve the problem ofthe
22
infrared divergence connected w ith the singularity of the e ective potentialat the origin. Indeed,C olem an and W einberg have show n that such m echanism really works in the nonsupersym m etric m assless
4
theory w ith U (1) gauge interaction in their originalpaper.
23
R EFER EN C ES [1]L.Ibanez and G .G .R oss,Phys.Lett.B 110,215 (1982);K .Inoue,A .K akuto,H .K om atsu and S. Takeshita, Prog. T heor. Phys, 68, 927 (1982); L. A lvarez-G aum e, M . C laudson and M .B.W ise,N ucl.Phys B 207,96 (1982). [2]N .Seiberg,Phys.Lett.B 318,469 (1993);N .Seiberg,Phys.R ev.D 49,6857 (1994); N .Intriligator,R .G .Leigh,and N .Seiberg,Phys.R ev.D 50,6857 (1994). [3]M .D ine and A .E.N elson,Phys.R ev.D 48,1277 (1993);M .D ine,A .E.N elson,and Y . Shirm an,Phys.R ev.D 51,1362 (1995);M .D ine,A .E.N elson,Y .N ir,and Y .Shirm an, Phys.R ev.D 53,2658 (1996). [4]I.A eck,M .D ine and N .Seiberg,Phys.Lett.B 137,187 (1984);H .M urayam a,Phys. Lett.B 355,187 (1995);N .K itazawa,N ucl.Phys.B 479,336 (1996). [5]Izawa K .-I.and T .Yanagida,Prog.T heor.Phys.95,949 (1996);K .Intriligator and S. T hom as,N ucl.Phys.B 473,121 (1996). [6]M .J. Strassler, Phys. Lett. B 376, 119 (1996); A .E. N elson and M .J. Strassler, hepph/9607362; [7]L.O ’R aifeartaigh,N ucl.Phys.B 96,331 (1975). [8]S.C olem an and E.W einberg,Phys.R ev.D 7,1888 (1973). [9]M .T .G risaru,W .Siegeland M .R ocek,N ucl.Phys.B 159,429 (1979). [10]Y .Shirm an,Phys.Lett.B 389,287 (1996). [11]L.R andall,hep-ph/9612426. [12]R eview ofParticle Properties,Phys.R ev.D 54,1 (1996).
24
FIG U R ES FIG .1. T he 1-loop diagram for the soft breaking m ass of the hypercolor gaugino. T he solid and dotted internallines denote the propagators of
Vi
and Vi (i= 1;2),respectively. T he crosses
denote the insertions ofF and m for the internalboson and ferm ion lines,respectively. FIG .2. T he 2-loop diagram s for the soft breaking m asses ofthe scalar preons. T he solid and dotted internal lines denote the propagators of
Vi
and Vi (i = 1;2), respectively. T he dashed
internal line denotes the propagator of the scalar preon P~ or N~. T he wavy line denotes the hypercolor gauge boson propagator. T he propagator ofthe hypercolor gaugino is denoted by the wavy line w ith the solid line. T he crosses denote the insertions ofF . FIG .3. T he 3-loop diagram for the soft breaking m asses ofthe standard m odelgauginos. T he solid and dotted internallines denote the propagators of the preon
P
and the scalar preon P~,
respectively. T he internalline ofthe hypercolor gaugino isdenoted by the wavy line w ith the solid line. T he dot on the propagator ofthe hypercolor gaugino denotes the 1-loop diagram in Fig.1. FIG .4. T he 1-loop diagram for the soft breaking m ass of the dow n-type H iggs boson. T he dotted internalline denotes the propagators ofD ,H ,D ,and H . It is not needed to consider the corrections ofthe scalar quarks and leptons,because their contributions cancelout each other.
FIG .5. T he 1-loop diagram to generate the soft breaking term B H H + h:c: in the H iggs potential. T he solid internallines denote the propagators ofthe H iggsinos H~ or H~ . T he wavy line w ith the solid line denotes the propagator ofthe SU (2)L gaugino or the U (1)Y gaugino. Since the contribution ofthe U (1)Y gaugino loop is far sm aller than that ofthe SU (2)L gaugino,it can be neglected in eq.(21). FIG .6. T he four-ferm ion interaction caused by the colored H iggs exchange. D denotes the colored H iggs. FIG .7. T he four-ferm ion interactions caused by the colored H iggsino exchange. D~ denotesthe ~L is also possible. colored H iggsino. T he diagram by replacem ent dL $ uL and ~ bL ! t
25
F V1
s s
A As A A
s
s s
s
s
s
H
V2
s
s A A A A
s
s
m V1
V2
Fig.1
26
H
Fig.2
@
Aq q q qAq qA A qq q A q q q q
(a)
(b)
(c)
A Aq qq A qq q q A q q q q q q q q q q q q q qA q q Aq A A
A q q qA Aq Aq qA A qq q A A q q q q q q q q q q qq qq q
(d-1)
(d-2)
@ @ @
A p p pAp pp p A p p A p p p p p p p p p pp pppp
@ @
+
A p p pAp pp p A p p A p p p p p p p p p pp pppp
27
q qq qq q
q
q q q q q
q q q q q q A A q Aq q qAq q A A A A
(d-3)
+
pAp p pp A pp p p p p p p p p p p p pp pAp p p A
+
pppp pp p p p p p p p p p p p p Ap p pApp p A A
s
s s
P
s
P~ s s
s
~
N
s
N s s s s s
Fig.3
28
D ;H ;D ;H r
r r r
r
r
r
r r r r r
r r r r r r
r r
H
r r r
r
Hy
Fig.4
@@ @@
H~ H
@@ @@
2; 1
Fig.5
29
H~ H
uL
dL HH
HH
D HH
HH
eR ; R ; R
uR Fig.6
~ bL
dL
uL
HH
HH
D~
3 HH
uR
HH
~ t R Fig.7 30
eR ; R ; R
