Table Of ContentDPNU-98-01
January 1998
Confining phase in SUSY SO(12) gauge theory with one
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spinor and several vectors
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Nobuhito Maru
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Department of Physics
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Nagoya University
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1 Nagoya 464-0814, JAPAN
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8 [email protected]
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X Abstract
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We study the confining phase structure of N=1 supersymmetric SO(12) gauge theory
with N ≤ 7 vectors and one spinor. The explicit form of low energy superpotentials
f
for N ≤ 7 are derived after gauge invariant operators relevant in the effective theory
f
are identified via gauge symmetry breaking pattern. The resulting confining phase
structure is analogous to N ≤ N + 1 SUSY QCD. Finally, we conclude with some
f c
comments on the search for duals to N ≥ 8 SO(12) theory.
f
PACS: 11.15.Tk, 11.30.Pb, 12.60.Jv
Keywords: Supersymmetric Gauge Theory, N = 1, SO(12), Confining Phase
1 Introduction
Ourunderstanding ofnon-perturbativenatureinN = 1supersymmetric gaugetheories
has much progressed since the pioneering works of Seiberg and his collaborators [1, 2].
Especially, physics of confining phase in N = 1 supersymmetric gauge theories was
enriched (quantum deformed moduli space, “s-confinement”). These works have been
also extended to the theories with various types of gauge groups and matter contents
[3, 4, 5, 6]. Furthermore, these theories have recently been applied to construction of
models with dynamical supersymmetry breaking or SUSY composite models.
In this paper, we study the confining phase in N = 1 supersymmetric SO(12)
gauge theory with N ≤ 7 vectors and one spinor. There are two motivations we are
f
interested in this particular model. First, from the theoretical point of view, it will
provide useful informations for finding the dual to SO(N )(N > 10) with an arbi-
c c
trary number of vectors and spinors. Although the duality of this class of models has
only been generalized to SO(10) [7], the known dualities have the following remarkable
properties which are not contained in Seiberg’s duality [13]: 1. Chiral-Nonchiral dual-
ity. 2. Reducibility to the exceptional group (G ) duality. 3. Simple and Semi-simple
2
group duality (without “deconfinement”). 4. Identification of massive spinors and Z
2
monopoles under duality [7, 8, 9, 10, 11, 12, 14]. Therefore, it is natural to ask whether
these properties exist in the duality for SO(N )(N > 10) theory. However, looking
c c
for this dual seems to be highly non-trivial from the result of Ref. [7]. Cho [4] has
already investigated in detail the confining phase of SO(11) gauge theory with N ≤ 6
f
vectors and a spinor and extracted some clues in search for duals. It is interesting
enough to pursue further following the line of his argument in order to clarify the dual
to SO(N )(N > 10) theory. Second, as mentioned in the above paragraph, the theory
c c
under consideration may provide phenomenologically viable models with dynamical
supersymmetry breaking or SUSY composite models.
This paper is organized as follows. In section 2, gauge invariant operators relevant
to the low energy physics are identified. We derive the explicit form of low energy
superpotentials for N ≤ 7 in section 3. In the last section, summary and some
f
comments on the search for duals to N ≥ 8 SO(12) theory are given.
f
1
2 The SO(12) model
The model we consider has following symmetry groups
G = SO(12) ×[SU(N ) ×U(1) ×U(1) ×U(1) ] (1)
gauge f V V Q R global
under which the superfields transform as1
Vi ∼ (12, ,1,0,0), (2)
µ
Q ∼ (32,1,0,1,0) (3)
α
and no tree level superpotential. Note that since each of the U(1) symmetries in Eq.(1)
are anomalous, the action is transformed as
g2
S → S −iCα d4x FF˜, (4)
32π2
Z
whereC denotestheanomalycoefficientofthecorrespondingU(1) SO(12)2,U(1) SO(12)2
V Q
or U(1) SO(12)2 anomalies and α is a transformation parameter. If the theta param-
R
eter in the Lagrangian is shifted under these anomalous U(1)’s as θ → θ + Cα, then
anomalies can be cancelled. Recalling the relation
Λ b0 8π2
= exp − +iθ , (5)
µ! g2(µ) !
where b represents 1-loop beta function coefficient 2
0
1
b = [3µ(Adj)− µ(R)] = 26−N , (6)
0 f
2
matter
X
andΛisthestrongcouplingscaleofthetheory,thespurionsuperfieldΛb0 istransformed
as
Λb0 ∼ (1,1,2N ,8,12−2N ). (7)
f f
Using these symmetries and holomorphy, we can easily fix the form of the dynami-
caly generated superpotential W for the small value of N so that U(1) charge of
dyn f R
W be 2 and U(1) ,U(1) charges vanish. The results are summarized in Table 1.
dyn V Q
N = 6 case is special because R charge of Λb0 vanishes, therefore we cannot construct
f
1We implicitly regard the 32 dimensional SO(12) spinor as the projection Q= P−Q64 where Q64
means the 64 dimensional spinor of SO(13) and P− = 12(1−Γ13).
2µdenotesquadraticDynkinindexdefinedasTrTa(R)Tb(R)=µ(R)δab (Ta: thegeneratorsofthe
group, R: representation which the superfield belongs to). We use the following values : µ(12) = 2,
µ(32)=8, µ(66)=20.
2
N R(Λb0) W
f dyn
0 12 (Λ26/Q8)1/6
1 10 (Λ25/Q8V2)1/5
2 8 (Λ24/Q8V4)1/4
3 6 (Λ23/Q8V6)1/3
4 4 (Λ22/Q8V8)1/2
5 2 Λ21/Q8V10
6 0 X(Q8V12 −Λ20)
7 −2 Q8V14/Λ19
Table 1: The form of dynamically generated superpotentials
the dyamically generated superpotential. However, the classical constraint among mat-
ter superfields is modified by non-perturbative effects and this quantum constraint can
be included in the superpotential by using the Lagrange mutiplier superfield X [1].
Therefore N = 6 case is analogous to N = N SUSY QCD (quantum deformation of
f f c
moduli space). Furthermore, N = 7 case is analogous to N = N +1 SUSY QCD (“s-
f f c
confinement”) [1] and N ≤ 5 case is analogous to N ≤ N −1 SUSY QCD (runaway
f f c
superpotential) [15].
In order to describe the low energy effective theory, we need to find gauge invariant
operators which behave as the moduli space coordinate 3. It is in general troublesome
to do this task. However, if the gauge symmetry breaking pattern is known at generic
points in the moduli space, one can easily identify these gauge invariant operators.
We illustrate below how it works in the present model. The gauge symmetry breaking
pattern we utilize is [17]
<32> <12> <12>
SO(12) −→ SU(6) −→ SU(5) −→ SU(4)
<−1→2> SU(3) <−1→2> SU(2) <−1→2> 1. (8)
Withthisinformationinhand,countingdegreesoffreedomofgaugeinvariantoperators
is nothing but a group theoretical exercise. We display in Table 2 parton degrees
of freedom, unbroken subgroups, eaten degrees of freedom by Higgs mechanism and
hadron degrees of freedom. We are now in a position to construct gauge invariant
operators explicitly. Before doing this, we need to notice that SO(12) spinor product
3Vacuum expectation values (VEV’s) of these gauge invariant operators are in one to one core-
spondence to the solutions of D-flatness conditions [16]
3
N Parton Unbroken Eaten Hadron
f
DOF Subgroup DOF DOF
0 32 SU(6) 66−35 = 31 1
1 44 SU(5) 66−24 = 42 2
2 56 SU(4) 66−15 = 51 5
3 68 SU(3) 66−8 = 58 10
4 80 SU(2) 66−3 = 63 17
5 92 1 66 26
6 104 1 66 38
7 116 1 66 50
Table 2: Degrees of freedom of independent gauge invariants
decomposes into the following irreducible representations
˜
32×32 = [0] +[2] +[4] +[6] (9)
A S A S
where [n] represents rank-n antisymmetric tensor, subscripts “A” and “S” mean anti-
symmetry and symmetry under spinor exchange, and the tilde of the last term implies
that the rank-6tensor isself-dual. Since our modelhas onlyonespinor, gaugeinvariant
˜
operators can include [2] and [6] in Eq.(9).
S S
Taking this into account, we can construct gauge invariant composites as follows4
1
L = (QTΓ[µΓν]CQ)(QTΓ Γ CQ) ∼ (1;1;4N ;4R),
[µ ν] f
2!2!
M(ij) = (Viµ)TVj ∼ (1; ;−8;2R),
µ
1
N[ij] = QTV/ iV/ jCQ ∼ (1; ;2N −8;4R),
f
2
1
P[ijklmn] = QTV/ [iV/ jV/ kV/ lV/ mV/ n]CQ ∼ (1; ;2N −24;8R),
f
6!
1
R[ijklmn] = ǫµ1···µ12(QTΓ ΓνCQ)(QTΓ Γ ···Γ CQ)Vi Vj VkVl Vm Vn
6! µ1 ν µ2 µ6 µ7 µ8 µ9 µ10 µ11 µ12
∼ (1; ;4N −24;10R), (10)
f
where the square bracket means the antisymmetrization of the corresponding indices
and V/ i ≡ ViΓµ and we use here the following SO(12) Gamma matrices,
µ
4The representations and the charge in Eq.(10) are those under Eq.(11)
4
Γ = σ ⊗σ ⊗σ ⊗σ ⊗σ ⊗σ Γ = −σ ⊗σ ⊗σ ⊗σ ⊗σ ⊗σ
1 2 3 3 3 3 3 2 1 3 3 3 3 3
Γ = 1⊗σ ⊗σ ⊗σ ⊗σ ⊗σ Γ = −1⊗σ ⊗σ ⊗σ ⊗σ ⊗σ
3 2 3 3 3 3 4 1 3 3 3 3
Γ = 1⊗1⊗σ ⊗σ ⊗σ ⊗σ Γ = −1⊗1⊗σ ⊗σ ⊗σ ⊗σ
5 2 3 3 3 6 1 3 3 3
Γ = 1⊗1⊗1⊗σ ⊗σ ⊗σ Γ = −1⊗1⊗1⊗σ ⊗σ ⊗σ
7 2 3 3 8 1 3 3
Γ = 1⊗1⊗1⊗1⊗σ ⊗σ Γ = −1⊗1⊗1⊗1⊗σ ⊗σ
9 2 3 10 1 3
Γ = 1⊗1⊗1⊗1⊗1⊗σ Γ = −1⊗1⊗1⊗1⊗1⊗σ
11 2 12 1
Γ = σ ⊗σ ⊗σ ⊗σ ⊗σ ⊗σ
13 3 3 3 3 3 3
and C is a charge conjugation matrix.
In order to see that these gauge invariant operators are in fact the coodinates of
the moduli space, one has to check whether the total degrees of freedom of these gauge
invariant operators in eq (10) coincide with hadronic degrees of freedom. In Table 3,
N Hadron DOF L M N P R Constraints
f
0 1 1 0 0 0 0
1 2 1 1 0 0 0
2 5 1 3 1 0 0
3 10 1 6 3 0 0
4 17 1 10 6 0 0
5 26 1 15 10 0 0
6 38 1 21 15 1 1 -1
7 50 1 28 21 7 7 -14
Table 3: Hadron degree of freedom count
these degrees of freedom are listed as a function of N . For N ≤ 5, hadronic degrees
f f
of freedom and that of L,M,N,P and R agree with each other. For N = 6, degrees
f
of freedom of L,M,N,P and R are larger than those of hadrons by one. This implies
that L,M,N,P and R are not independent and a single constraint among them exists.
This statement isalso consistent with theprevious argument fordynamically generated
superpotentials. For N = 7, 14 constraints are expected to come from the equations
f
of motion as in N = N +1 SUSY QCD.
f c
We can obtain more non-trivial support which convinces us that gauge invariant
operators L,M,N,P and R are moduli. One of the powerful methods to study the low
energy spectrum is ’t Hooft anomaly matching [18]. To see that anomalies between
elementary fields and composite ones match, we take anomaly free symmetry group
instead of Eq.(1)
G = SO(12) ×[SU(N )×U(1)×U(1) ] (11)
AF gauge f R global
5
where new U(1) and U(1) are linear combinations of the original U(1)’s in Eq.(1).
R
Matter superfields transform under Eq.(11) as
Vi ∼ (12, ,−4,R), (12)
µ
Q ∼ (32,1,N ,R) (13)
α f
where R = N −6/N +4. Wecancalculate anomaliesandsee thatanomaliesmatchfor
f f
N = 7; SU(7)3 : 12A( ),SU(7)2U(1) : −120µ( ),SU(7)2U(1) : −48µ( ),U(1) :
f R 11 R
−434,U(1) : −112,U(1)2U(1) : 2128,U(1) U(1)2 : −29120,U(1)3 : −28154,U(1)3 : 5600,
11 R 121 R 11 R 1331
where A( ) and µ( ) are cubic and quadratic Dynkin indices for fundamental
representation of SU(N ). Recalling that anomalies are saturated in N = N + 1
f f c
SUSY QCD, this coindence for N = 7 is very natural and gives a strong support that
f
the theory in this case is in “s-confinement” phase.
For N ≥ 8, it is impossible to satisfy anomaly matching conditions without violat-
f
ing N ≤ 7 result even if other gauge invariant operators are added. This implies that
f
a confining phase of this model terminates at N = 7.
f
3 Low energy superpotentials
Inthissection, wedetermineexplicitlylowenergysuperpotenialsintermsofL,M,N,P
and R. Since we know which gauge invariant operators are moduli in the previous sec-
tion, it is straightforward to work out what should be in the superpotential. Following
Ref. [4], we first determine the quantum deformed constraint in N = 6 theory. Di-
f
mensional analysis, symmetries and holomorphy restrict the superpotential as follows
W = X(R2 +P2L+2PPfN +L2detM
Nf=6
1
+ ǫ ǫ LNi1j1Ni2j2Mi3j3Mi4j4Mi5j5Mi6j6
2!4! i1i2i3i4i5i6 j1j2j3j4j5j6
1
+ ǫ ǫ Ni1j1Ni2j2Ni3j3Ni4j4Mi5j5Mi6j6 −Λ20) (14)
4!2! i1i2i3i4i5i6 j1j2j3j4j5j6 6
where Λ is the strong coupling scale of SO(12) gauge theory with 6 vector flavors and
6
a spinor. Coefficients of each terms are determined so that it reproduce the superpo-
tential in SO(11) gauge theory with 5 flavors [4].(If VEV < V6 >6= 0 is given, SO(12)
12
gauge theory with 6 flavors under consideration here reduces to SO(11) gauge theory
with 5 flavors)
One may also determine these coefficients by using the symmetry breaking along
the spinor flat direction SO(12) <−3→2> SU(6),explicitly < 32 >T= (0,a,0,···,0,a,0).
6
Vi decomposes into 6+¯6 under SU(6), which is explicitly as
µ
q +q¯
1 1
i(q −q¯ )
1 1
q +q¯
2 2
i(q −q¯ )
2 2
q3 +q¯3
i(q −q¯ )
V = 3 3 (15)
µ q +q¯
4 4
i(q −q¯ )
4 4
q +q¯
5 5
i(q5 −q¯5)
q +q¯
6 6
i(q −q¯ )
6 6
where q ,q¯(i = 1,···,6) mean SU(6) quarks, antiquarks, respectively. The reduced
i i
theory is SU(6) gauge theory with 6( + ), therefore it has a single quantum
constraint [1].
According to this decompostion rule, SO(12) gauge invariant operators are decom-
posed into the following SU(6) meson mij, baryon b and anti-baryon ¯b;
L → 12a4
Mij → 2(mij +mji)
N[ij] → −4ia2(mij −mji)
P → 64ia2(b+¯b)+2ia2ǫ mijmklmmn
ijklmn
R → 64a4(b−¯b) (16)
Using this information, one can also determine coefficients so that detm−b¯b = Λ12 be
reproduced.
The superpotential of N = 7 case can be found in a similar way. In this case,
f
the superpotential must have the following features. 1. It is smooth everywhere on
the moduli space. 2. Equations of motion give classical constraints among vectors
and a spinor. 3. Adding mass term for one vector flavor to this superpotential and
integrating out this massive vector, the superpotential (14) must be reproduced. The
result is5
1
W = (MijR R −2iNijP R +LP P Mij
Nf=7 Λ19 i j i j i j
5Although this superpotential has already been derived in Ref. [3] by using the index argument,a
PMN3 term was missing. Without this term, the result of Ref. [4] cannot be correctly recovered.
7
1
+L2detM + 3!22ǫi1···i7PjMji1Ni2i3Ni4i5Ni6i7
1
+ ǫ ǫ LNi1j1Ni2j2Mi3j3Mi4j4Mi5j5Mi6j6Mi7j7
3!4! i1i2i3i4i5i6i7 j1j2j3j4j5j6j7
1
+ ǫ ǫ Ni1j1Ni2j2Ni3j3Ni4j4Mi5j5Mi6j6Mi7j7).(17)
4!3! i1i2i3i4i5i6i7 j1j2j3j4j5j6j7
By adding the mass terms for vector fields δW = m Mij to the superpotential
ij
(14) and integrating out each massive vectors successively, we can readily derive the
superpotentials for N ≤ 5 systematically. As a matter of fact, we obtain
f
1
W = Λ21/(L2detM + LNi1j1Ni2j2Mi3j3Mi4j4Mi5j5ǫ ǫ
Nf=5 5 2!3! i1i2i3i4i5 j1j2j3j4j5
1
+ Ni1j1Ni2j2Ni3j3Ni4j4Mi5j5ǫ ǫ ),
4! i1i2i3i4i5 j1j2j3j4j5
Λ22 1/2
W = 2 4 ,
Nf=4 L2detM + 2!12!LNi1j1Ni2j2Mi3j3Mi4j4ǫi1i2i3i4ǫj1j2j3j4 +(PfN)2!
Λ23 1/3
W = 3 3 ,
Nf=3 L2detM + 21!LMi1j1Ni2j2Ni3j3ǫi1i2i3ǫj1j2j3!
Λ24 1/4
W = 4 2 ,
Nf=2 L2detM +LN2!
Λ25 1/5
W = 5 1 ,
Nf=1 L2M!
Λ26 1/6
W = 6 0 , (18)
Nf=0 L2 !
where the strong coupling scales for each flavor are related to each other through one-
loop matching of gauge coupling as follows
Λ26 = m Λ25 = m m Λ24 = m m m Λ23
0 11 1 11 22 2 11 22 33 3
= m m m m Λ22 = m m m m m Λ21
11 22 33 44 4 11 22 33 44 55 5
= m m m m m m Λ20 = m m m m m m m Λ19. (19)
11 22 33 44 55 66 6 11 22 33 44 55 66 77 7
It is worth to note that one can confirm the above superpotentials (18) to recover
correctly the superpotentials for N ≤ 4 in SO(11) theory [4], which is obtained when
f
one flavor vector field has non-vanishing VEV.
Before closing this section, we briefly discuss dynamical supersymmetry breaking.
For N = 6, if we take the tree level superpotential as
f
W = λ S V2 +λ S Q2V6 +λ S Q4V6 (20)
tree 1 1 2 2 3 3
8
where S are singlet superfields, then equations of motion with respect to S
1,2,3 1,2,3
and the quantum constraint (14) are incompatible. Therefore, supersymmetry is dy-
namically broken [19]. For N = 0, since we cannot add terms which lift a classical
f
flat direction (i.e.L) preserving U(1) to the tree level superpotential, supersymmetry
R
remains unbroken. The same argument seems to be applicable for 1 ≤ N ≤ 5 6.
f
4 Summary
In this paper, we have studied the confining phase in N = 1 SO(12) SUSY gauge
theory with N ≤ 7 vectors and a spinor. Utilizing the gauge symmetry breaking
f
pattern at generic points on the moduli space which plays a crucial role in our study,
we have identified gauge invariant operators which behave as the moduli coordinate.
Then we have derived explicitly low energy superpotentials for N ≤ 7.
f
Some clues in search for duals are obtained from the results of this work. Let us
suppose that the dual with the gauge group G˜ exists for N ≥ 8. The original SO(12)
f
theory breaks down to SU(6)+N ( + ) along the spinor flat direction. On the other
f
hand, the gauge group of the dual theory is usually unbroken since in the dual theory
the gauge invariant operators develop VEV. Since SU(6)+N ( + )(N ≥ 8) theory
f f
is dual to SU(N −6)+N ( + ) [13], we can guess that at least G˜ must include
f f
SU(N −6) as a subgroup to preserve the duality along this direction. Furthermore,
f
the superpotential in the dual theory must recover N = 7 superpotential. We also
f
note that N = 8 case in our model is known to be self-dual [20].
f
Although it seems to be quite difficult to find a dual which is compatible with the
above requirements, we hope that this work will provide useful informations to search
for the dual to SO(N )(N ≥ 11) theory.
c c
Acknowlegdements
The author thanks S. Kitakado and T. Matsuoka for careful reading of the manuscript
and for useful discussions.
6In Ref. [3], the authors construct a model with dynamical supersymmetry breaking for N = 1
f
by promoting a global U(1) to a local U(1) and adding singlets to cancel U(1) gauge anomaly.
9
