Download Dynamical Generation of the Gauge Hierarchy in SUSY

Progress of Theoretical Physics Supplement No. 123, 1996
431
Dynamical Generation of the Gauge Hierarchy in SUSY-GUT's
Tomohiro HOTTA, Ken-Iti IZAWA and Tsutomu YANAGIDA*)
Department of Physics, University of Tokyo, Tokyo 113, Japan
(Received November 17, 1995)
We make a review of the recently proposed supersymmetric hypercolor gauge theory with
six flavors of quarks interacting strongly at the grand unification scale, where dynamical
breaking of the grand unified SU(5)GuT produces a massless pair of composite color-triplet
states. Yukawa interactions between elementary Higgs multiplets and the hyperquarks yield
a pair of massless SU(2}L doublets, giving large masses to the color-triplet partners. We
prove that this pair of massless Higgs doublets survives quantum corrections and remains
in the low-energy spectrum as far as the supersymmetry is unbroken. Hence this model
realizes naturally the large mass hierarchy in the grand unified theories. We also show that
the dangerous dimension-five operators for nucleon decays are suppressed simultaneously.
§1.
Introduction
The pair of light Higgs doublets is an inevitable ingredient in the supersymmetric
(SUSY) standard model. In the SUSY grand unified theories (GUT's), however, one
must require an extremely precise adjustment of parameters in order to obtain a pair
of light Higgs doublets. l) Therefore, the necessity of the fine-tuning of parameters
seems to be a crucial drawback in the SUSY-GUT's, although the recent high-precision
measurements on the Weinberg angle have shown a remarkable agreement 2 ) with the
prediction of the SUSY extension l) of the GUT. 3 )
There have been, in fact, several attempts 4 )- 8) to have the light Higgs doublets
without requiring the fine-tuning of parameters. In Ref. 8) it has been shown that the
dynamics of SUSY QCD-like theory at the GUT scale naturally generates pairs of Higgs
doublets at low energies. However, quantum effects have not been fully investigated in
Ref. 8) and seven flavors of (hyper)quarks have been assumed in order to have a manifest
symmetry to protect massless Higgs doublets from quantum corrections. Although
this model is very interesting due to its natural accommodation of the Peccei-Quinn
symmetry, the low-energy spectrum is rather involved with two pairs of light Higgs
doublets.
In the recent work 9 ) we have investigated the quantum dynamics of this QCD-like
theory and shown that a pair of massless Higgs doublets in the minimal model with
six (hyper)quarks is indeed stable quantum mechanically. This makes the dynamical
approach proposed in Ref. 8) yet more attractive.
§2.
Classical vacua in the Higgs phase
Our model is based on a supersymmetric hypercolor SU(3)H X U(l)H gauge theory8l with N1 flavors of quarks Q~ in the 3 representation and antiquarks QA in the 3*
representation of SU(3)H, where o: = 1, · · ·, 3 and A= 1, · · ·, N1. The chiral multiplets
•) Talk is given by T. Yanagida.
T. Hotta, K.-1. Izawa and T. Yanagida
432
have U ( 1) H charges +1 and -1, respectively.
Before investigating the realistic case Nt = 6, 9 ) we first consider the basic case
Nt = 5. The anomaly-free flavor symmetry is then given by
Q~ and
QA
SU(5)L x SU(5)R
X
U(1)v
X
U(1)R,
(1)
under which the quark multiplets transform as
Q~
:
QA:
( 5, 1' +1'
g) '
(1,5,--1,g)
(2)
(a= 1, · · ·, 3; A= 1, · · ·, 5)
We note that U(1)v is nothing other than U(1)H, but we regard it as a flavor group.
The GUT gauge group SU(5)auT is also embedded in a part of the flavor group SU(5)L
X SU(5)R· Namely, the quarks Q~ and QA transform as 5* and 5 under SU(5)auT,
respectively.
When SU(5)auT is spontaneously broken down to the standard gauge group, there
appear unwanted Nambu-Goldstone multiplets. To avoid them, we introduce a chiral
multiplet EAB in the adjoint (24) representation of SU(5)GUT· B) Then, we have a
superpotential
(3)
3
where m and mE denote mass parameters. Here, we have omitted a Tr(E ) term in
the superpotential for simplicity, since the presence of this term does not change the
conclusions in this review.
The classical vacua satisfy the following equations:
(4)
where
(5)
Together with the D-term flatness condition for the gauge group SU(5)auT x SU(3)H
x U ( 1) H, we find four distinct vacua. Since three of them are not interesting phenomenologically, we restrict our discussion to one vacuum given by (up to gauge and
Dynamical Generation of the Gauge Hierarchy in BUSY-GUT's
433
global rotations)
0
0
v
0
0
Q~=
0
0
0
v
0
0
0
0
0
v
Qa' A-
(
0
0
0
v
0
0
0
0
0
0
v
0
),
0
0
v
(6)
3
2
EAB= m
),
3
2
_
-1
v-
~~mmE
2
),2
0
-1
-1
The vacuum-expectation values in Eq. (6) break the original gauge group down to the
standard gauge group:
SU(5)cvT
X
SU(3)H x U(l)H ~ SU(3)c x SU(2)L x U(1)y.
(7)
Here, U(1)y is a linear combination of U(1)H and a U(1) subgroup of SU(5)cUT·
The GUT unification of three gauge coupling constants of SU(3)c x SU(2)L x U(1)y
is achieved in the limit giH -+ oo, where giH is the gauge coupling constant of the
hypercolor U(1)H· s) If one requires the GUT unification by 2% accuracy, one gets a
constraint a1H 2': 0.06 for as ~ 1/25 at the GUT scale (see Ref. 8) for the normalization
of O'.Jfl).IO)
It is amusing that there is no fiat direction and hence no massless state, which
results from the fact that the superpotential (3) breaks explicitly the flavor symmetry
(1) down to SU(5)cvT x U(1)n. Nambu-Goldstone multiplets transforming as (3*, 2)
and (3, 2) under SU(3)c x SU(2)L are absorbed in the SU(5)cvT gauge multiplets to
form massive vector multiplets.
Let us now turn to the Nt = 6 case with an additional pair of quark Q~ and
antiquark Q6, whose mass term is forbidden by imposing an axial U(1)A symmetry
(8)
These massless quarks have fiat directions since they do not have a superpotential with
EA8 and hence there are infinitely degenerate vacua. Around the vacuum in Eq. (6),
the fiat directions are given by
Q~=(O,O,w),
Q6=(
~
weio
)·
(9)
When w = 0, the standard gauge group SU(3)c x SU(2)L x U(1)y remains unbroken.
We assume that SUSY-breaking effects choose it as a true vacuum, avoiding w =I= 0
where the color SU(3)c would be broken.
It is remarkable that there is a pair of massless color-triplets Q~ and Q6 in the above
vacuum at w = 0. Notice that the unbroken color SU(3)c is a diagonal subgroup of
the hypercolor SU(3)H and an SU(3) subgroup of SU(5)auT and the original SU(3)H
T. Hotta, K.-1. Izawa and T. Yanagida
434
quarks Q~ and Q~ transform as 3 and 3* under the color SU(3)c. The presence of
massless color-triplets Q~ and Q~ is a crucial point for obtaining a pair of light Higgs
doublets as seen in §4.
Before proceeding to the quantum analysis of the vacuum chosen above, two remarks are in order here:
i) The classical moduli space of vacua has a hypercolor-gauge invariant description
in terms of the observable "meson" M and "baryons" B and B:
A
_
A -a
M B - Q<>QB,
B[ABCJ = }:_Eaf3-yQAQBQC
3!
<> f3 'Y '
-
B[ABC]
=
1
(10)
-a-(3--y
j!Eaf3-yQAQBQC.
The vacua corresponding to Eqs. (6) and (9) are given by
0
0
(11)
B[3,4,5]
= 13[3,4,5] = v3'
with all the other components of B and fJ vanishing.
It is clear that the flavor gauge group SU(5)auT x U(1)H is spontaneously broken
down to SU(3)c x SU(2)L x U(1)y in the w = 0 vacuum. Then we have a pair of
massless composite states
(12)
which are 3 and 3* of the color SU(3)c, respectively.
ii) We note that there is another interesting vacuum in the present model:
Q~=
0
0
0
0
0
0
0
0
0
v
0
0
0
0
0
0
v
0
'
QaA-
(
0
0
0
0
0
0
0
0
0
0
v
0
0
0
v
0
0
0
),
(13)
2
3
EAB=
m
>.
2
3
_V~mmE
3
2
3
v-1
-1
),2
.
Dynamical Generation of the Pauge Hierarchy in SUSY-GUT's
435
In this vacuum U(1)y is a diagonal subgroup of U(1)'s in SU(5)auT and SU(3)H
and hence the additional U(1)H is not necessary. However, unwanted massless states
Q~ and Q~ exist in addition to a pair of SU(2)L doublets Q~ and Q6 (a= 2, 3) which
may be identified with the Higgs multiplets in the standard model. These massless multiplets Q~ and Q~ have U(1)y charges -1 and +1, respectively, and give an additional
contribution to the renormalization-group equations for gauge coupling constants. This
destroys the success of gauge coupling unification in the SUSY-GUT. Thus we do not
investigate this vacuum in this review. ll)
§3.
Quantum vacua in the confining phase
Let us analyze quantum effects on the classical vacua given in Eqs. (6) and (9).
We see that the effective mass matrix for quarks has four zero eigenvalues in our vacua:
~m
2
A
A
m
I
0
m =
(14)
0
0
0
0
Therefore, our model becomes a SUSY QCD-like theory with the effective N1 = 4 at
low energies. 9 )
Expanding the E fields around the values (E) given in Eq. (6), we write the
tree-level superpotential as
W
=Qf (~ m 8ij + ACTij) Q~ + >. Q~ (CTab) Q~ + >. Q~ (CTai) Q~ + >. Qf (CTia) Q~
>. CTo ( 3Qi
-a i
+ J30
Qa -
1 mE {Tr( CT 2) + CT 2} + -J30
mmE CTo ,
2Q-aQa)
a a + 2
- -->.0
2
(i,j=1,2; a,b=3,···,5)
(15)
where CT and CTo are defined by
3
E
A
B
A
= (E B)+ CT
A
B
+
- 11
v30
3
-2
CTo,
-2
-2
(16)
(A, B = 1, · · ·, 5)
Notice that there is a tadpole term for CTo in Eq. (15), which is canceled out by the
Q~ Q~ condensation.
We now think of our model as the QCD-like theory with two massive (Q~ and Qi)
and four massless ( Q~ and Q~ ; a = 3, · · · , 6) quarks interacting with the 0' fields. Then
T. Hotta, K.-I. lzawa and T. Yanagida
436
we can integrate out the massive quarks and irrelevant a fields 12 ) to get the low-energy
effective theory described by four massless quarks with the superpotential
Wiow
=
A
Q~ ( aab -
vk
ao 8ab)
Q~
+21 mE { Tr(a 2 ) + ao 2 } + -V30
2
(17)
mmE
--A-ao .
(a,b=3,···,5)
Here we have eliminated the N ambu-Goldstone modes stemming from breakdown of
the flavor symmetry, since they will be absorbed in the gauge multiplets of SU(5)GUT
to form massive vector multiplets.
We next proceed to find out the effective superpotential described by the composite
meson M and baryons B and B interacting with the a fields. It is highly nontrivial to
obtain the superpotentials dynamically generated by strong interactions. However, it
has become clear recently that the effective superpotentials can be exactly determined
for certain classes of SUSY nonabelian gauge theories. 15 ), 16 ) In particular, for the
QCD-like theory with Nt = Nc + 1 flavors of quarks where Nc is the number of colors,
the quantum moduli space of vacua is the same as the classical one and the effective
superpotential at low energies is uniquely determined. 16 )
According to Refs. 15) and 16), we obtain the effective superpotential for our
model from Eq. (17) as
=
Wdyn
(18)
A-
5 ( BaMab[Jb
- det M),
where Mab , Ba and tJa are composite meson and baryon states in the Nt
=4
case:
Mab "' Q~ Qb,
Ba
"'
Ba,....,
1\
2
af3'Y
( 3 !)
(~)
3!
E
Eabcd
2
E
Ot/3"(
Qb Qc Qd
a {3 'Y'
EabcdQ-OtQ-f3Q-"~
b c d
(19)
•
(a b = 3 · · · 6)
'
'
'
A denotes the dynamical scale of SU(3)H, and
a is defined by
1
1
(20)
From Eq. (18) we find that the quantum vacua satisfy
Mab[Jb
=
8 det M
oM%
BaMab = 0,
- 0
+ B a [Jb + As A-a ba -
(21)
Dynamical Generation of the Gauge Hierarchy in BUSY-GUT's
437
for a, b = 3, · · · , 6;
(22)
for a, b = 3, · · ·, 5. Using the D-term flatness condition for the flavor gauge group
SU{5)auT X U(1)H together with Eqs. {21) and {22), we obtain the quantum vacua
which are exactly the same as those given in Eq. {11) for the classical theory. We take,
as before, the SU(3)c x SU(2)L x U(1)y invariant vacuum at w = 0.
In this vacuum, the mass matrix for color-triplet states M, B and f3 is given by
(23)
Notice that M and B (B) have canonical dimensions two and three, respectively. A
pair of massless color-triplet states is now a mixture of M, Band f3 with its orthogonal
states having GUT scale masses. We will see later that the structure of the mass matrix
for these color-triplet states is crucial for suppression of the dangerous dimension-five
operators for nucleon decays.
The Nambu-Goldstone modes in Mab (a, b = 3, · · ·, 5) get tree-level masses through
the interactions {the>. coupling term) with a and ao given in Eq. {18). Thus there are
no other massless composite states besides the SU(3)c triplets observed in the mass
matrix (23).
§4.
Generation of the light Higgs doublets
We introduce a standard Higgs multiplets, HA and fiA (with A= 1, · · ·, 5), in the
fundamental representations 5 and 5* of SU(5)GUT· The mass term for them is forbidden by the axial U{1)A symmetry given in Eq. (8). Assuming their transformation
property under U ( 1) A as
(24)
we have a superpotential
(25)
These Yukawa interactions induce the following mass terms at low energies:
{26)
for a= 3, · · ·, 5. Thus the color-triplet components of HA and fiA denoted by Ha and
[Ia (a= 3, · · ·, 5) become massive together with the massless composites in M 6a, Ma6,
Ba and [Ja. On the other hand, the doublet components of HA and fiA denoted by Hi
and [Ii (i = 1, 2) remain massless.
T. Hotta, K.-I. Izawa and T. Yanagida
438
This missing partner mechanism is easily understood in terms of the Higgs phase,
where Q~ and Q~ have vacuum-expectation values for a= 3, · · ·, 5 as shown in Eq. (6).
Then, it is clear from Eq. (25) that the color-triplets Ha and jja get masses together
with Q6 and Q~, respectively. On the other hand, the SU(2)L-doublet Higgses Hi and
jji have no partners to form masses and the Hifii mass term itself is forbidden by the
axial U(1)A in Eq. (24). However, we need a more careful analysis at the quantum
level, since the axial U(1)A is broken by the hypercolor SU(3)H instantons.
We now prove that the Higgs doublets Hi and jji are exactly massless even at the
quantum level in the limit of SUSY being exact. We first integrate out the massive
quarks Q~ and Qf to get the low-energy superpotential (neglecting the irrelevant C1
fields)
(27)
where Wiow is given in Eq. (17). Using the methods proposed in Refs. 15) and 16), we
obtain the effective superpotential
1
I
Wetr = Wetr
2hh
i
6
+ hHaM a6 + h 1H- aM 6a + --HiH
M 6 ,
5m
-
(28)
where Weff is given in Eq. 18). Since M 66 vanishes in the present vacuum, no mass
term for the Higgs doublets Hi and jji is generated.
This important conclusion is also understood if one notices that the hypercoloranomaly (instanton) effects are independent of any Yukawa coupling constants>., hand
h 1 , and hence they are present only in the dynamical part Wdyn in W~tr· Therefore, the
Yukawa-coupling dependent parts of W~ff must be U(1)A-invariant, which shows that
the Hifii terms are always accompanied by M 66 as seen in Eq. (28). On the other
hand, the dynamical part Wdyn does not have the Hifii term. This is easily proved by
means of a higher symmetry in the limit of h = h 1 = 0 (e.g., Hi-t eio: Hi, jji - t eif3 jji
and all the other fields intact).
Note that the sixth quarks will become massive when the Higgs doublets Hi and
jji acquire the vacuum expectation values at the electroweak scale, which is consistent
with the choice w = 0 in Eq. (11).
§5.
Discussion
We have shown in this review that our QCD-like theory with six quark flavors generates one pair of massless Higgs doublets naturally. 9 ) These Higgs doublets survive all
the quantum corrections and remain in the massless spectrum. The masslessness of the
original Higgs multiplets, HA and fiA (.A= 1, · · ·, 5), is understood by the axial U(1)A
symmetry. Although this axial U(1)A is broken by the hypercolor instantons, the presence of massless Higgs doublets has been proved by means of the nonrenormalization
theorem in SUSY theories. 15 ), 16 )
It is remarkable that nucleon decays due to the dangerous dimension-five operators
are suppressed in our model. These operators are induced by exchanges of SU(3)c
triplet Higgs multiplets Ha and fia. 17 ) Owing to (23) and (26), the mass matrix for
Dynamical Generation of the Gauge Hierarchy in SUSY-GUT's
439
the color-triplet sector is given by
(29)
where
0
fnc
=
h'
0
h
0
v4
v3
As
As
v3
v2
A5
A5
(30)
The dimension-five operators are proportional to ( fn(J 1 ) , which is none other than
11
zero. This conclusion can also be understood in terms of the Higgs phase: the mass for
Q~ and Q6 is not generated even in the presence of instanton effects and hence there
is no transition matrix between Ha and fia. We note that this is consistent with the
result by Affieck, Dine and Seiberg. 18 )
The GUT unification of three gauge coupling constants of SU(3)c x SU(2)L x U(l)y
is achieved in the strong coupling limit of U(l)y. The necessity of the strong U(l)n
is a possible drawback in the present model. However, it is very hard to distinguish
phenomenologically the standard SUSY-GUT and our model, since the SU(3)c gauge
coupling constant has experimental errors of several percent (see Ref. 8)). Moreover,
some threshold corrections from GUT -scale particles also give a few percent ambiguity
to the renormalization-group equations. 19 )
From a theoretical point of view, it may be a problem that the electromagnetic
charge quantization is not an automatic consequence in our model. However, this can
be solved by assuming the hypercolor SU(3)n x U(l)n to be embedded in, for example,
SU(4)n at some higher scale. This extension of the hypercolor group also solves the
problem that the gauge coupling constant of U(l)n blows up below the Planck scale.
This possibility together with a phenomenological analysis will be considered in a future
communication. 20 )
In the present review we have assumed the global SUSY theory. In the framework of
supergravity, the flat directions (wand wei 8 ) in Eq. {9) may be lifted substantially when
they reach the Planck scale. However, effects from such vacuum shifts are negligibly
small near the origin (w = 0) of the flat directions.
In our model the global U(l)A plays a central role for having a pair of massless
Higgs doublets. Such a global symmetry may be, in general, broken by topologychanging wormhole effects. 21 ) If it is the case, the pair of Higgs doublets is no longer
massless. The magnitudes of these induced operators are, however, not known and
hence we assume that these effects are sufficiently suppressed.
It should be pointed out here that there is an alternative model to generate light
Higgs doublets in the present framework. If we assume SU(2)n as the hypercolor
gauge group instead of SU(3)n, the strong hypercolor dynamics produces directly a
massless pair of composite SU(2)L doublets, Q~Qi and Q~Q6 (i = 1, 2), as noted in
the previous paper. 8 ) These bound states may be identified with the standard-model
Higgs doublets Hi and fii, respectively. In this model, however, we need to assume
T. Hotta, K.-I. Izawa and T. Yanagida
440
the following nonrenormalizable interactions to have effective Yukawa couplings to the
ordinary quarks and leptons, 5* and 10;
·10QAQa
JL5*
a 6
M
·10Q 6a QaA·
+ L10
M
(31)
With M being the gravitational scale M ~ 2.4 x 10 18 GeV we obtain effective Yukawa
couplings of order of v / M ~ 10- 2 . This is too small to explain the observed mass for
the top quark. A possible solution to this problem is given by introducing a pair of
massive Higgs H~ (5) and JI'A(5*) and consider the superpotential,
J' 10 ·10H~ + f" Q~QA_JI'A + mHH~JI'A.
scale and f' "' f" "' 0(1), we obtain 0(1)
W =
(32)
effective Yukawa
For mH at the GUT
coupling of the composite Higgs ,...., Q~Qi to the top quark.
Finally we make a comment on the basic structure of our model. We have assumed
a mass term for the first five quarks as seen in Eq. (3). However, we can obtain the
same result by introducing a coupling with a singlet field cp instead of the mass term.
In such a model the vacuum-expectation value of cp plays a role of the mass m. This
model seems very intriguing, since it may be regarded as a low-energy effective theory
of a QCD-like theory dual to the present model where there are no gauge singlets cp
and E. 14 ) This intriguing possibility will be discussed in the forthcoming paper. 20 )
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14
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Dynamical Generation of the Gauge Hierarchy in BUSY-GUT's
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