Nonabelian Duality and Higgs Multiplets in Supersymmetric Grand Unified Theories

16 Pages·0.17 MB·English

by  T. Hotta

UT-738
Nonabelian Duality and Higgs Multiplets
in Supersymmetric Grand Unified Theories
6
T. Hotta, Izawa K.-I. and T. Yanagida
9
9
1
Department of Physics, University of Tokyo
n
a Bunkyo-ku, Tokyo 113, Japan
J

4
2 February 1, 2008

1
v
0
Abstract
2
3
We consider strongly interacting supersymmetric gauge theories which break dy-
1
0 namically the GUT symmetry and produce the light Higgs doublets naturally. Two
6 models we proposed in the previous articles are reanalyzed as two phases of one
9
theory and are shown to have desired features. Furthermore, employing nonabelian
/
h
duality proposed recently by Seiberg, we study the dual theory of the above one and
p
- show that the low-energy physics of the original and dual models are the same as
p
expected. We note that the Higgs multiplets in the original model are regarded as
e
h composite states of the elementary hyperquarks in its dual theory. Theories with
:
v other hypercolors and similar matter contents are also analyzed in the same way.
i
X
r
a
1
1 Introduction
The supersymmetric grand unified theory (SUSY-GUT) [1] is one of the promising can-
didates for the physics beyond the standard model. In fact, the recent high-precision
measurements on the standard-model parameters such as the Weinberg angle agree with
some of its predictions [2]. In spite of the remarkable success, there is a fatal fault in the
SUSY-GUT: a fine-tuning problem. Since the GUT scale, typically 1016 GeV, is extremely
high compared with the weak scale ∼ 102 GeV, we have to adjust parameters in the GUT
accurately in order to have a Higgs doublet in the standard model. Although a number of
attempts have been made to solve this serious problem, there was no convincing model to
explain the origin of the light Higgs doublet.
In recent papers [3, 4, 5] we have proposed SUSY gauge theories whose interactions are
strong at the GUT scale causing dynamical breaking of the GUT symmetry. These models
also provide mechanisms which produce the light Higgs doublet naturally.
The main purpose of this paper is to examine the strongly interacting SUSY gauge
theories more thoroughly. In addition to the method used to find quantum vacua in Ref.[4,
5], we employ nonabelian duality which has been proposed recently by Seiberg [6] as a
powerful tool to investigate nonabelian gauge theories. Since the nonabelian duality states
that SU(N ) and SU(N −N ) gauge theories with the common N flavors have the same
c f c f
low-energy behavior, especially on the vacuum structure, we may reduce the number of
theories to study and also check consistency of the results using both theories.
In section 2 we review the results of our two models in Ref.[4, 5] and show that these
models are regarded as two phases of one theory. In section 3 we consider the dual theory of
the model in section 2 and show that the low-energy physics of the dual model is the same
asthat ofthe originalone. We, however, stress that short-distance structures of theoriginal
and dual models are different from each other and hence these models represent different
physics above the GUT scale. We also note that the Higgs multiplets in the original model
are composite states of the elementary hyperquarks in the dual theory. In section 4 we
extend our analysis to theories with other hypercolors and similar matter contents. Section
5 is devoted to our conclusions. We also comment on some extensions and modifications
of the models.
2
2 The original model
We review the models studied in Ref.[4, 5] in which light Higgs doublets are generated
dynamically. We analyze these two models in a unified manner treating them as different
phases of a single theory.
The model is based on a supersymmetric hypercolor SU(3) gauge theory with six
H
flavors of hyperquark chiral superfields QA and Q¯α (α = 1,···,3;A = 1,···,6) in the
α A
fundamental representations 3 and 3∗ of SU(3) , respectively. The first five QI and Q¯α
H α I
(I = 1,···,5) transform as 5∗ and 5 under the GUT gauge group SU(5) , respectively,
GUT
while the last Q6 and Q¯α are singlets of SU(5) . We also introduce three kinds of
α 6 GUT
SU(3) -singlet chiral superfields: a pair of H and H¯I, ΣI and Φ (I,J = 1,···,5) which
H I J
are 5, 5∗, 24+1 and 1 of SU(5) .
GUT
We impose a global U(1) symmetry:
A
QI,Q¯α → QI,Q¯α,
α I α I
Q6,Q¯α → eiξQ6,eiξQ¯α,
α 6 α 6
H ,H¯I → e−iξH ,e−iξH¯I,
I I (1)
ΣI → ΣI ,
J J
Φ → e−2iξΦ,
(I = 1,···,5),
to forbid such terms as H¯IH and Q¯αQ6 in the superpotential. Then, the superpotential
I 6 α
is given by1
W = λΣI Q¯αQJ +hH Q¯αQI +h′H¯IQ¯αQ6 +fΦQ¯αQ6
J I α I 6 α I α 6 α
1 1 (2)
+ m Tr(Σ2)+ m′ (TrΣ)2 −µ TrΣ.
2 Σ 2 Σ Σ
Here, we have omitted trilinear self-coupling terms of Σ for simplicity since they are irrel-
evant to the conclusion. The global U(1) has a strong SU(3) anomaly and hence it is
A H
broken by instanton effects at the quantum level. However, as shown in Ref.[4] the broken
global U(1) even plays a crucial role to protect a pair of massless Higgs doublets from
A
having a mass.
1We have chosen the normalization of the singlet field TrΣ so that the Yukawa term among ΣI , Q¯α
J I
and QI is written with a single coupling constant λ as shown in Eq.(2). The effect of the rescaling of the
α
field TrΣ appears in the K¨ahler potential, but it is irrelevant to the present analysis.
3
Let us first consider a classical vacuum discussed in Ref.[4]:
0 0 0
0 0 0
  0 0 v 0 0 0
v 0 0
hQAi =  , hQ¯αi = 0 0 0 v 0 0 ,
α  0 v 0  A  
  0 0 0 0 v 0
 0 0 v   
   
 0 0 0 
  (3)
 
1
1
µ  
hΣI i = Σ 0 ,hH i = hH¯Ii = 0,
J m +2m′   I
Σ Σ  0 
 
 
 0 
 
 
where
m µ
v = Σ Σ . (4)
λ(m +2m′ )
s Σ Σ
Here, the vacuum-expectation value of Φ is undetermined since its potential is flat for
hQ6i = hQ¯αi = 0. In this classical vacuum the gauge group is broken down as
α 6
SU(3) ×SU(5) → SU(3) ×SU(2) . (5)
H GUT C L
There is no unbroken U(1) , and we introduce an extra U(1) gaugesymmetry in Ref.[3, 4]
Y H
to have the standard-model gauge group unbroken below the GUT scale v.2
Remarkable is that the missing partner mechanism [7] does work very naturally in this
classical vacuum [3]. Namely, the color triplets H and H¯I (I = 3,···,5) acquire the GUT-
I
scale masses together with Q6 and Q¯α, respectively. On the other hand the SU(2) -doublet
α 6 L
H and H¯I (I = 1,2) remain massless, since they have no partners to form massive chiral
I
superfields with.
We now discuss quantum vacua where vacuum-expectation values of the Higgs fields
ΣI , H and H¯I take forms given in Eq.(3).
J I
In these vacua two hyperquarks QI and Q¯α (I = 1,2) become massive. The integration
α I
of the two massive hyperquarks leads to a low-energy effective theory having the other
four massless hyperquarks QI and Q¯α (I = 3,···,6). We can then express the effective
α I
2The GUT unificationofthree gaugecouplingconstants inthe standardmodelis realizedin the strong
coupling limit of U(1) [3].
H
4
superpotential with meson Mi , baryon B and antibaryon B¯i chiral superfields
j i
Mi ∼ Qi Q¯α,
j α j
B ∼ ǫαβγǫ QjQkQl ,
i ijkl α β γ (6)
B¯i ∼ ǫ ǫijklQ¯αQ¯βQ¯γ,
αβγ j k l
as follows [5]:
W = Λ−5(B Mi B¯j −detMi )+λΣa Mb
eff i j j b a
+hH Ma +h′H¯aM6 +fΦM6
a 6 a 6
hh′
− (H¯1H +H¯2H )M6 (7)
1 2 6
m
1 1 m m′ m µ
+ m Σa Σb + Σ Σ (Σa )2 − Σ Σ Σa ,
2 Σ b a 2 m +2m′ a m +2m′ a
Σ Σ Σ Σ
λµ
where Λ denotes a dynamical scale of the low-energy SU(3) interactions, m = Σ ,
H m +2m′
Σ Σ
a,b = 3,···,5 and i,j,k,l = 3,···,6. This superpotential implies a flat direction satisfying
Λ−5(B B¯6 −detMa )+fΦ = 0.
6 b
Let us consider, among the vacua of Eq.(7), the two vacua which satisfy hΦi = 0 or
hB i = hB¯6i = 0.
6
Vacuum (a): The vacuum with hΦi = 0 is analyzed in Ref.[4].3 We see that B and B¯6
6
have non-vanishing vacuum-expectation values leading to breaking of the U(1) subgroup
Y
of SU(5) . Thus we need to introduce an extra U(1) gauge symmetry so as to have
GUT H
the standard-model gauge group unbroken below the GUT scale.
Notice that this quantum vacuum is the same as the classical one, which is consistent
with the fact that the classical moduli space is not altered by quantum corrections for the
case of N = N + 1 [8] where N and N are the numbers of flavors and colors of the
f c f c
massless hyperquarks, respectively.
Vacuum (b): The vacuum with hB i = hB¯6i = 0 is analyzed in Ref.[5]. That is
6
hB i = hB¯ii = 0,
i
hM6 i = hMa i = hM6 i = 0,
a 6 6 (8)
3
m µ 1 m µ
hMa i = Σ Σ δa , hΦi = Σ Σ ,
b λ(mΣ +2m′Σ) b fΛ5 "λ(mΣ +2m′Σ)#
where the GUT gauge group is broken down to the standard-model one, namely SU(3) ×
C
SU(2) × U(1) . Thus, there is no need to introduce an extra U(1) , differently from
L Y H
3
Although the model in Ref.[4] does not contain the singlets TrΣ and Φ, the vacuum considered in
Ref.[4] is equivalent to that with hΦi=0 in the present model.
5
the previous phase (a). An interesting point is that this quantum vacuum differs from the
classical one which satisfies B B¯6 −detMa = 0. This result agrees with the conclusion in
6 b
Ref.[8] for the case of N = N . Notice that the effective N is three (= N ) in the present
f c f c
phase since the vacuum-expectation value of Φ induces a mass for Q6 and Q¯α.
α 6
As noted in Ref.[5], we have a pair of massless bound states B and B¯6 in this vacuum.
6
Since they have non-vanishing U(1) charges, they contribute to the renormalization-group
Y
equations of three gauge coupling constants in the standard model. A change of running
of couplings threatens to destroy the GUT unification of gauge coupling constants which
is regarded as one of the motivations for considering the SUSY-GUT as a unified theory.
However, itseemsquitereasonabletoassumethattherearenonrenormalizableoperators
in the superpotential suppressed by some scale M higher than the GUT scale (originating
0
fromgravitationalinteractions,forexample). Amongsuchoperatorsweconsiderthelowest-
dimensional nonrenormalizable operator consistent with our gauge and global symmetries
which is to contain baryon superfields. That is
′
δW = Mf3ǫαβγǫα′β′γ′(QIαQJβQKγ )(Q¯αI′Q¯βJ′Q¯γK′). (9)
0
This interaction generates a mass term for B and B¯6 in the effective superpotential as
6
f′
δW = B B¯6, (10)
eff M3 6
0
which corresponds to the physical mass for B and B¯6
6
f′Λ4
m ≃ . (11)
B6 M3
0
If one takes M in Eq.(11) at the gravitational scale, i.e. M ≃ 2 × 1018 GeV, and
0 0
Λ ≃ 3×1016 GeV, for example, one has the mass for B and B¯6 ∼ 1011 GeV for f′ ∼ O(1).
6
This mass is too small compared with the GUT scale and the presence of B and B¯6
6
destroys the GUT unification of gauge coupling constants. However, since M is given
0
at the gravitational scale, it evolves as the change of scale by renormalization effects.
Provided that the renormalized MR becomes about 1017 GeV at the GUT scale,4 we
0
4Above the GUT scalethe number ofeffectively masslessflavorsis six andthe presentmodel liesin the
conformalwindow [6]. Therefore,the model has a quasi-infraredfixed point. As pointed out in Ref.[9] the
renormalization factor Z for the wave functions of quarks Q and Q¯ goes to vanish in the long distance.
Q
ThissuggeststherenormalizedmassM0R ≡ZQ2M0 becomessmallerastherenormalizationpointislowered.
We also suspect that this kind of renormalization effects may be an origin of the GUT scale itself.
6
obtain m ∼ 1015 GeV for Λ ≃ 3×1016 GeV and f′ ∼ O(1). This result turns out to be
B6
consistent with the recent experimental data on the three gauge coupling constants [5].
In both the phases (a) and (b), the colored Higgs H and H¯a acquire masses of the
a
GUT scale with the composite states M6 and Ma from the interactions in Eq.(7), but the
a 6
Higgs doublets H and H¯I (I = 1,2) remain massless because of hM6 i = 0.
I 6
It is remarkable that the operator in Eq.(9) changes the quantum moduli space. Actu-
ally, the vacuum (a) is no longer in the quantum moduli space, while the vacuum (b) still
remains there. Since it is quite natural to consider that the operators suppressed by the
gravitational scale exist, there is a doubt as to the presence of the vacuum (a).
3 The dual model
Nonabelian duality proposed in Ref.[6] enables us to study supersymmetric nonabelian
gauge theories with different color groups. The duality means that an SU(N ) gauge theory
c
withN flavorsofquarks andanSU(N −N )gaugetheorywiththesamenumber ofquarks
f f c
have the same behavior in the infrared limit, especially the same moduli space of vacua.
Thus, one may investigate the dual theory instead of our original model, which may even
clarify the structure of the original model. Furthermore, there is a possibility that we can
throw a new light on the origin of the fields in the original model, such as H and H¯I, as
I
composite fields of the elementary hyperquarks in the dual theory.
Now let us consider the nonabelian dual of the model in section 2, which is expected to
coincide with the original model as a low-energy effective theory.
The dual gauge group turns out to be SU(3), and thus we denote it as SU(3) in
H˜
distinction with the original gauge group SU(3) . The dual model is described by a su-
H
persymmetric SU(3) gauge theory with singlet chiral superfields σA and six flavors of
H˜ B
dual hyperquarks q¯α and qA in the fundamental representations 3∗ and 3, respectively. In
A α
addition we have to introduce Yukawa couplings between hyperquarks and σA to obtain
B
a correct global symmetry and correct vacua.
Then the dual superpotential to Eq.(2) is given by
W˜ = λ˜q¯ασA qB +λρΣI σJ +hρH σI +h′ρH¯Iσ6 +fρΦσ6
A B α J I I 6 I 6
1 1
+ m Tr(Σ2)+ m′ (TrΣ)2 −µ TrΣ, (12)
2 Σ 2 Σ Σ
(I,J = 1,···,5),
7
where ρdenotestheduality scaleto matchtheoperatordimensions ina correspondence [10]
ρσA ∼ QAQ¯α , (A,B = 1,···6). (13)
B α B
To obtainEq.(12) thehyperquarks QA andQ¯α inEq.(2) aresubstituted withσA by means
α A B
of the relation Eq.(13).
The fields in Eq.(12) transform under a global U(1) symmetry as
A
q¯α,qI → q¯α,qI,
I α I α
q¯α,q6 → e−iξq¯α,e−iξq6,
6 α 6 α
σI → σI , (14)
J J
σI ,σ6 → eiξσI ,eiξσ6
6 I 6 I
σ6 → e2iξσ6 ,
6 6
with the transformation law in Eq.(1).
Since the dual superpotential in Eq.(12) looks complicated, we reduce it by integrating
out the superfields H , H¯I, Σ, Φ, σI , σ6 and σ6 to
I 6 I 6
1 1
W˜ ′ = λ˜q¯ασI qJ + m Tr(σ2)+ m′ (Trσ)2 −µ Trσ, (15)
I J α 2 σ 2 σ σ
where
(λρ)2 (λρ)2m′ λρµ
m = − , m′ = Σ , µ = − Σ . (16)
σ m σ (m +5m′ )m σ m +5m′
Σ Σ Σ Σ Σ Σ
Notice that the sixth hyperquarks q¯α and q6 are massless. If we integrate out σI , Eq.(15)
6 α J
becomes a superpotential including only q¯α and qI with nonrenormalizable interactions
I α
1 1
(q¯αqJ)2 and (q¯αqJ)2. From this viewpoint, the superfields σI are interpreted as
m I α m′ I α J
σ σ
composite states of q¯α and qI.
J α
Let us study vacua corresponding to the original vacua given in the previous section
which satisfy
1
1
µ  
hσI i = σ 1 . (17)
J m +3m′  
σ σ  0 
 
 
 0 
 
 
The integration of massive hyperquarks q¯α and qI (I = 1,2,3) leads to a low-energy
I α
effective theory. Since this effective theory is an SU(3) gauge theory with three massless
H˜
hyperquarks q¯α and qA (A = 4,5,6), its effective superpotential is obtained by means of
A α
8
knowledge for the N = N case in Ref.[8]. The effective superpotential is described by
f c
meson Mi , baryon B and antibaryon B¯ chiral superfields,
j
Mi ∼ qiq¯α,
j α j
B ∼ ǫαβγq4q5q6,
α β γ (18)
B¯ ∼ ǫ q¯αq¯βq¯γ,
αβγ 4 5 6
as follows:
W˜ = X(BB¯ −detMi −Λ˜6)+λ˜σb Ma
eff j a b
1 1 m m′ m µ (19)
+ m σa σb + σ σ (σa )2 − σ σ σa ,
2 σ b a 2 m +3m′ a m +3m′ a
σ σ σ σ
where Λ˜ denotes a dynamical scale of the low-energy SU(3) interactions, a,b = 4,5, and
H˜
i,j = 4,5,6. This superpotential implies a flat direction satisfying BB¯−detMi −Λ˜6 = 0.
j
We consider two vacua, hM6 i = 0 and hBi = hB¯i = 0, which have a pair of massless
6
Higgs doublets. Here these Higgs doublets are all composite bound states of the dual
hyperquarks M6 ∼ q6q¯α and Mi ∼ qiq¯α (i = 4,5).
i α i 6 α 6
Vacuum (a’): The vacuum with hM6 i = 0 corresponds to the vacuum (a) in the
6
previous section, where the fields have the following vacuum-expectation values:
hBi = hB¯i = Λ˜3,
m µ
hMa i = − σ σ δa ,
b λ˜(m +3m′ ) b (20)
σ σ
hM6 i = hMa i = 0, hσa i = 0.
a 6 b
In this vacuum the SU(5) breaks down to SU(3) × SU(2) since the baryon B and the
GUT C L
antibaryon B¯ have non-vanishing U(1) charges. Therefore, we need an extra U(1) as in
Y H
the vacuum (a). Interesting enough, although the point hBi = hB¯i = 0 and detMi = 0
j
with unbroken U(1) is in the classical moduli space, this point disappears from the moduli
Y
space by non-perturbative effects at the quantum level. This phenomenon contrasts with
the vacuum (b) in section 2 in which the U(1) breaks down classically but is restored
Y
quantum mechanically.
Although there is no problem phenomenologically if the U(1) is strong enough at
H
the GUT scale, the U(1) brings some theoretical problems. First of all, the U(1) is
H H
not asymptotically free and its gauge coupling constant blows up at some higher scale.5
Secondly, the charge quantization is left unexplained.
5For m ∼ 1018 GeV, the mass of the SU(2) triplet in Ma is of the order of Λ˜2/m which may be
σ L b σ
smaller than the GUT scale. In this case, the unification of the three gauge coupling constants is realized
within the experimentalerrors,evenif the gauge coupling g of U(1) is not so large. Thus, it is possible
H H
that the coupling g does not diverge below the Planck scale.
H
9
Vacuum (b’): The vacuum with hBi = hB¯i = 0 corresponds to the vacuum (b) in the
original model. We find the vacuum:
λ˜2(m +3m′ )2
hM6 i = − σ σ , (21)
6
m2µ2
σ σ
and the other fields acquire the same expectation values as in the vacuum (a’) which breaks
SU(5) down to SU(3) × SU(2) × U(1) . Therefore, we do not need to introduce an
GUT C L Y
extra U(1) . As in (a’) this quantum vacuum is different from the classical one which
H
satisfies detMi = 0.
j
As in the vacuum (b), there is a pair of massless baryon B and antibaryon B¯ with
non-vanishing U(1) charges. Unlike in the original model, however, nonrenormalizable
Y
operators generating the mass for B and B¯ are forbidden by the global U(1) symmetry
A
in Eq.(14). The effective superpotential in Eq.(19) says that nonperturbative effects never
generate the mass term of the baryon B andthe antibaryon B¯ although the U(1) is broken
A
by instanton effects. Thus, there remains a pair of the baryons in the massless spectrum,
whichrendersthevacuum(b’)unrealistic. Italsosuggeststhattheshort-distancebehaviors
of the original and dual models are different.6 On the other hand, any nonrenormalizable
operators with the U(1) symmetry do not affect the stability of the vacuum (a’). Indeed,
A
it is rather guaranteed by an introduction of the term Φ˜M6 which leads to hM6 i = 0.
6 6
Wenoteaninteresting relationbetween theoriginalandthedualmodels. Intheoriginal
model the Higgs multiplets are regarded as elementary fields. Whereas in the dual model
the Higgs doublets (even including σI ) are composite states of the dual hyperquarks q¯α
J A
and qA. Moreover, all the Higgs multiplets, H , H¯I, ΣI and Φ, in the original model might
α I J
be regarded as composite states of the elementary hyperquarks in the dual model.
To summarize, the dual model reproduces exactly the same low-energy physics as the
originalmodeldoes. This supports thecorrectness ofthe duality arguments. Weproceed to
use this nonabelian duality as a powerful tool to investigate supersymmetric gauge theories
with other hypercolors in the next section.
6Ifoneaddsanonrenormalizableoperator Mf˜′3q4q5q6q¯4q¯5q¯6,onereproducesthesamelow-energyphysics
0
as in the previous vacuum (b). However, short-distance physics are different from each other, since in the
original model the global U(1) is unbroken at the classical level whereas there is no such a symmetry in
A
its dual model.
10