Proton Decay Constraints on Low Scale AdS/CFT Unification
James B. Dent∗ and Thomas W. Kephart†
Department of Physics and Astronomy,
Vanderbilt University, Nashville, TN 37235
(Dated: October 30, 2018)
Dark matter candidates and proton decay in a class of models based on the
AdS/CFT correspondence are discussed. We show that the present bound on the
proton decay lifetime is inconsistent with N = 1 SUSY, and strongly constrains
N = 0 non-SUSY, low scale trinification type unification of orbifolded AdS⊗S5
models.
∗Electronic address: james.b.dent@vanderbilt.edu
†Electronic address: tom.kephart@gmail.com
http://arxiv.org/abs/0704.1451v1
mailto:james.b.dent@vanderbilt.edu
mailto:tom.kephart@gmail.com
I. INTRODUCTION
Models with low scale (∼ TeV) unification are of great potential interest for LHC physics
since the unification scale, and thus any new particles that exist at that scale, will be well
within the reach of the LHC. Many models based on the AdS/CFT correspondence are of
this type. When orbifolded they can lead to products of SU(N) gauge groups[1], (Actually
they are products of U(N) gauge groups, and the extra U(1)s can be anomalous. However,
counter terms can be added the Lagrangian to cancel these anomalies. See [2] and references
therein. In what follows, we will suppress the U(1) factors, and assume they decouple from
the analysis.) with particle representations that can be identified with the Standard Model
(SM) when the gauge group G is broken to SU(3)× SU(2)× U(1). The simplest examples
are AdS ⊗ S5/Zn models with N = 3 so G = SU(3)
n where the matter fields are all in
bifundamental representations. ( For n = 3 this leads to a three family N = 1 trinification
model [3]. For a general classification of both N = 1 and N = 0 Zn orbifolded models see
[4, 5].) While some progress has been made [6, 7, 8], the full phenomenological implications
of these models are not known as detailed model building has been lacking. Recently,
however, some attempts have been made to accommodate a dark matter candidate [9], the
lightest conformal particle (LCP), and a stable proton [10] by the (well motivated yet ad hoc)
assignment of discrete charges to particles, which thereby forbids any harmful interactions.
Here we will investigate whether such assignments can arise naturally via discrete subgroups
upon spontaneous symmetry breaking, and determine their consequences.
We will carry out our analysis within a single representative model, and then show how
the results generalize. We begin with a review of the simplest non-SUSY low scale example
based on a Z7 orbifold [11]. We then follow the quantum numbers through the symmetry
breaking from G = SU(3)7 to the trinification [12] group SUC(3)× SUL(3)× SUR(3), then
to the group that contains B −L symmetry, SUC(3)× SUL(2)×UY (1)×UB−L(1)×UX(1),
and finally to the standard model. We show that a dark matter candidate LCP can only
arise if an appropriate R-parity assignments [9] is realized phenomenologically by adding
extra scalars in this model. We also show that dimension six proton decay operators are
present and therefore proton decay occurs much too rapidly in this model. This is always
the case for any low scale N = 1 SUSY trinification model of this type. There is a possible
escape in the non-SUSY case, but it will take clever model building to achieve an acceptable
result. The set of charge assignments that avoids proton decay given in [10] is such a case,
but it is not realized in the Z7 orbifold model studied here.
II. NON-SUSY Z7
We begin with a short review of the non-SUSY Z7 orbifold model [11], but before doing so
we will first pause to recall how models are constructed by orbifolding AdS ⊗ S5 [1]. When
we desire an N = 1 supersymmetric theory, we embed the orbifolding group Γ in the 4 of
the SU(4) R-symmetry of the underlying N = 4 theory via 4 = (a1, a2, a3, a4), where a1 = 1
and the other aj are nontrivial elements of Γ . We will only be concerned with abelian
Zn choices for Γ which generate chiral supermultiplet matter fields in the bifundamental
representations of the gauge group SUn(N) = SU1(N) × SU2(N) × ....SUn(N). When
Γ is Zn the aj are of the form aj = e
2πinj
n and the matter bifundamentals are (N, N̄)
representations of SUi(N) × SUi+nj(N) for all i = 1, 2, ...n and nj . (Note, n1 = 0 in the
N = 1 SUSY case.) If we wish to break all the supersymmetry, a1 must also be nontrivial,
and therefore n1 6= 0. The fermions are still generated by the embedding of Zn in the 4
of the initial SU(4) R-symmetry, but now the scalars arise from the embedding of Zn in
the 6 = (4 × 4)A, generated from the embedding of the 4. The scalar bifundementals are
in (N, N̄) representations of SUi(N) × SUi+pk(N), where the pk are obtained from the six
antisymmetric combinations e
2πipk
n = e
2πi(nj+n
n . We now have sufficient background to write
down the non-SUSY Z7 model on which we will focus most of our attention.
A. Initial Particle Content and sin2 θW
The fermions are given by the embedding of the Z7 orbifolding group in the 4 of the
SU(4) R symmetry: 4 = (α, α, α2, α3) [11] where α = e
7 , and the scalars can be found
from 6 = (α2, α3, α3, α4, α4, α5), which leads to the particle content
Fermions
2(3,3̄,1,1,1,1,1) (3,1,3̄,1,1,1,1) (3,1,1,3̄,1,1,1)
2(1,3,3̄,1,1,1,1) (1,3,1,3̄,1,1,1) (1,3,1,1,3̄,1,1)
2(1,1,3,3̄,1,1,1) (1,1,3,1,3̄,1,1) (1,1,3,1,1,3̄,1)
2(1,1,1,3,3̄,1,1) (1,1,1,3,1,3̄,1) (1,1,1,3,1,1,3̄)
2(1,1,1,1,3,3̄,3) (1,1,1,1,3,1,3̄) (3̄,1,1,1,3,1,1)
2(1,1,1,1,1,3,3̄) (3̄,1,1,1,1,3,1) (1,3̄,1,1,1,3,1)
2(3̄,1,1,1,1,1,3) (1,3̄,1,1,1,1,3) (1,1,3̄,1,1,1,3)
Scalars
(3,1,3̄,1,1,1,1) 2(3,1,1,3̄,1,1,1) 2(3,1,1,1,3̄,1,1) (3,1,1,1,1,3̄,1)
(1,3,1,3̄ 1,1,1) 2(1,3,1,1,3̄,1,1) 2(1,3,1,1,1,3̄,1) (1,3,1,1,1,1,3̄)
(1,1,3,1,3̄ 1,1) 2(1,1,3,1,1,3̄,1) 2(1,1,3,1,1,1,3̄) (3̄,1,3,1,1,1,1)
(1,1,1,3,1,3̄,1) 2(1,1,1,3,1,1,3̄) 2(3̄,1,1,3,1,1,1) (1,3̄,1,3,1,1,1)
(1,1,1,1,3,1,3̄) 2(3̄,1,1,1,3,1,1) 2(1,3̄,1,1,3,1,1) (1,1,3̄,1,3,1,1)
(3̄,1,1,1,1,3,1) 2(1,3̄,1,1,1,3,1) 2(1,1,3̄,1,1,3,1) (1,1,1,3̄,1,3,1)
(1,3̄,1,1,1,1,3) 2(1,1,3̄,1,1,1,3) 2(1,1,1,3̄,1,1,3) (1,1,1,1,3̄,1,3)
At the unification scale
sin2(θW ) =
where SUL(2) ⊂ SU
p(3), UY (1) ⊂ SU
p(3), and p and q are from the seven initial SU(3)s
[4]. We find this by starting with the gauge group SUn(3) and breaking to SUC(3) ×
SUL(3)× SUR(3) where SUC(3) ⊂ SU
r(3), SUL(3) ⊂ SU
p(3), and SUR(3) ⊂ SU
q(3) where
n = p + q + r, then writing Y as the sum of diag(1
,−1) of SUL(3) and diag(1,1,-2) of
SU(3)R we get
sin2θW =
3 + 5 (α2/αY )
3 + 5
at the unification scale. For the model at hand, p = 2 and q = 1, hence the result 2/7.
B. Symmetry Breaking
We proceed as in [13] by giving a VEV successively to (1,3,1,3̄,1,1,1), (1,1,3,3̄,1,1),
(1,1,3,3̄,1), and (1,1,3,3̄). Each VEV breaks an SU(3) × SU(3) pair to its diagonal SU(3)
subgroup and thus, after all four of these scalars have obtained VEVs, the gauge group
is SUC(3) × SUL(3) × SUR(3) with three massless chiral families of fermions in complex
representations
3[(3, 3̄, 1) + (1, 3, 3̄) + 3(3̄, 1, 3)]F
plus fermions in real representations that acquire heavy (unification scale) masses, as well
as various scalars given by
[19(1, 1, 1)+(1, 8, 1)+9(1, 1, 8)+3(3, 3̄, 1)+2(3̄, 3, 1)+8[(1, 3, 3̄)+(1, 3̄, 3)]+4(3̄, 1, 3)+3(3, 1, 3̄)]S
We can then break SUL(3) to SUL(2)×UL8(1) (the subscript “8” refers to the λ8 generator
of SUL(3)) by giving a VEV to the neutral singlet in the adjoint (1,8,1) which decomposes
as 8 = 10 + 23 + 2−3 + 30 under SUL(2) × UL8(1) (the 2s are the fundamental SUL(2)
representation and the subscript refers to the UL8(1) charge). We can also break SUR(3) to
UR3(1) × UR8(1) by first breaking to SUR(2) × UR8(1) as just described (except that the 8
is now chosen from one of the nine (1,1,8) multiplets) and then giving a VEV to a neutral
UR3(1) × UR8(1) singlet (which arises from the decomposition of an SUR(2) adjoint, which
is 30 = 12 + 1−2 + 10 under UR3(1)× UR8(1)).
The quantum numbers for hypercharge (Y ), baryon-minus-lepton (B−L), and X can be
written in terms of the U(1) charges T
L,R (i.e., the Cartan subalgebra charges) as follows
Y = −
T 8L +
T 8R −
T 3R (1)
B − L = −
T 8L −
T 3R (2)
X = −T 8L + 2T
R (3)
C. Low Energy Particle Content
The gauge group is now SUC(3)× SUL(2)× UY (1)× UB−L(1)× UX(1) and the particle
content is given by (listing only light fermions, but all the scalars):
Family Fermions
Q: 3(3, 2) 1
,1 L: 3(1, 2)− 1
,−1,3 ū: 3(3̄, 1)− 2
d̄: 3(3̄, 1) 1
,2 ē: 3(1, 1)1,1,0 N : 3(1, 1)0,1,0
Exotic Fermions
h̄: 3(3̄, 1) 1
,−4 h: 3(3, 1)− 1
Ē: 3(1, 2) 1
,0,−3 E: 3(1, 2)− 1
,0,−3 S: 3(1, 1)0,0,6
Scalars
8(1,1)1,0,0 8(1,1)0,1,0 8(1,1)0,1,−6 8(1,1)1,1,−6
8(1,1)−1,0,0 8(1,1)0,−1,0 8(1,1)0,−1,6 8(1,1)−1,−1,6
8(1,1)0,0,6 8(1,1)1,1,0 8(1,2) 1
,0,3 8(1,2) 1
,0,−3
8(1,1)0,0,−6 8(1,1)−1,−1,0 8(1,2)− 1
,0,−3 8(1,2)− 1
8(1,2) 1
,1,−3 3(3,1)− 1
,−2 3(3,1) 2
,−2 3(3,1)− 1
8(1,2)− 1
,−1,3 2(3̄,1)− 1
,−2 4(3̄,1)− 2
,2 4(3̄,1) 1
3(3,1)− 1
,−2 3(3,2) 1
,1 26(1,1)0,0,0
4(3̄,1) 1
,2 2(3̄,2)− 1
As in a usual trinified model, one can see that the fermion content corresponds to the
Standard Model (SM) particles plus a right handed neutrino, N , plus eleven additional
(exotic) particles.
III. GAUGED R-PARITY AND DARK MATTER CANDIDATES
Before discussing R-symmetry, we first dispense with U(1)X which we break with a VEV
for a (1, 1)0,0,6. In [9] it was shown that if a Z2 R-symmetry is imposed, then one has a
natural candidate for dark matter in the lightest conformal particle (LCP). Here we will
argue that this Z2 R-symmetry does not naturally arise in the present model and will not
generically be present in any trinified model which originates from orbifolded AdS5 × S
Our reasoning is similar to that found in [14] where criteria are given for a gauged R-parity
surviving in E6 (as well as Pati-Salam and SO(10)). The E6 case is similar to ours in that
the E6 sub-group examined is SUC(3)×SUL(2)×UY (1)×UB−L(1)×UX(1), and the fermions
are in a 27 of E6 with the same transformations as the fermion content in the model we are
examining.
The current model contains a continuous U(1)B−L symmetry which, when broken by a
scalar VEV carrying an even integer amount of 3(B−L), will result in a discrete (−1)3(B−L)
symmetry. This remaining discrete symmetry will then become the Z2 required in [9] and
one will be assured of an LCP dark matter candidate. One sees that only the scalars
(3, 1)−1/3,−2/3,−2, (3̄, 1)1/3,2/3,2, (3̄, 1)1/3,2/3,−4, and (3, 1)−1/3,−2/3,4 satisfy the criteria that if
one were to obtain a VEV then there would remain the desired discrete R-symmetry. Since
any such VEV would break SU(3)C , we conclude that the Z2 R-symmetry can not be
arranged in this model, since any other color singlet VEV breaks B − L completely.
IV. PROTON DECAY
In typical grand unified trinification models, unification occurs at MG ≃ 10
14GeV [15],
and rapid proton decay is avoided. Due to the low scale of unification that can arise in orb-
ifolded AdS/CFT models proton decay must be strictly forbidden by choices of field content
and interaction terms. A mechanism for accomplishing this via baryon charge assignment
was put forth in [10] where one can assign various baryon numbers to the scalar sector and
therefore exclude the unwanted interactions. We will examine if such a mechanism exists in
the present Z7 model.
Once scalar VEVs are obtained, quark and lepton masses are generated from the Yukawa
couplings terms
λq(QH1ū+QH2d̄) + λl(LH1ē+ LH2N) (4)
where we have defined the scalars H1: (1, 2)1/2,0,−3 and H2: (1, 2)−1/2,0,−3.
Additional interactions are seen to exist due to the presence of colored scalars in the
model. In particular there are the Yukawa terms
λc(QQβ + ēs̄β) (5)
involving the colored scalar that we label β which has charges (3,1)− 1
These are seen to be B and L nonconserving operators and will give rise to the interaction
shown in Figure 1. Once the scalars are integrated out, one will be left with a dimension
six effective four-fermion operator. This is disastrous for the model due to incredibly rapid
FIG. 1: Interaction leading to proton decay
proton decay via p → K+ + ν. The rate of proton decay due to this interaction is given by
Γ = A
where mβ is the mass of the scalar β. The dimensionful proportionality constant, A (of total
dimension five), contains relatively well known aspects, given by lattice QCD calculations,
as well as model dependent Yukawa couplings. In a typical trinified model [12], proton decay
(with reasonable couplings) was suppressed since colored triplet scalars acquire masses on
the order of the unification scale, 1014GeV. Here one would expect their masses to be on
the order of a few TeV and therefore one can immediately conclude that the low scale of
unification will produce a proton decay rate in gross disagreement with current experimental
bounds.
This has severe consequences for any AdS/CFT orbifold SUSY model with low-scale
unification due to the fact that these dangerous colored scalars will always be present in the
form of the superpartner of the fermion we have labeled h. This leads us to believe that if
a viable conformal model with low scale unification exists, it will not be supersymmetric.
V. CONCLUSIONS
We have examined the Z7 orbifold of AdS5 ⊗ S
5 which is the minimal low scale model
that can produce the SM fermion content with all three generations present. It was shown
that the scalar content is sufficient to produce the breaking pattern given by SU(3)7 →
SUC(3)×SUL(3)×SUR(3) → SUC(3)×SUL(2)×UY (1)×UB−L(1)×UX(1). After breaking
UB−L(1) and UX(1) it was shown that there exist two Higgs bosons that will provide the usual
Yukawa couplings to give the lepton and quark masses and break SUL(2)× UY (1) → UEM .
We also shown that there are no scalars in the spectrum to break the continuous UB−L(1)
to leave a Z2 R-symmetry which would have provided an LCP dark matter candidate. This
is in contrast to previous work where the Z2 symmetry was implemented by hand. Proton
decay was also examined in the model and found to be disastrous due to the presence of
colored scalars which mediates rapid proton decay due to their masses naturally occurring
at the TeV unification scale (as opposed to the usual high scale of unification of trinified
models which will provide agreement can the current bounds on proton decay). It was then
argued that this is an impasse for any supersymmetric models for this types of orbifolding,
since the rapid proton decay will be a feature of any generic low scale model that contains
SUSY. For the non-SUSY case, it still remains possible but challenging to find a model
without the problematic scalar fields.
As the Z7 is the minimal model from which one can obtain a low scale three generation
non-supersymmetric realization, evading the proton decay problem due to the existence of
colored scalars will be a steep challenge. As one increases the order of the Abelian orbifold
group, one will also increase the particle content and thus the dangerous colored scalars
seem to be ubiquitous in models that result in trinification. One possible avenue to explore
would be to examine models of type SU(N)n with minimal n, but with N not equal to 3
and determine if the problem persists.
Acknowledgments
We thank Paul Frampton for a useful discussion. This work was supported by U.S. DoE
grant number DE-FG05-85ER40226.
[1] A. E. Lawrence, N. Nekrasov and C. Vafa, Nucl. Phys. B 533, 199 (1998)
[arXiv:hep-th/9803015].
[2] E. Di Napoli and P. H. Frampton, Phys. Lett. B 638, 374 (2006) [arXiv:hep-th/0603065].
[3] S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998) [arXiv:hep-th/9802183].
[4] T. W. Kephart and H. Pas, Phys. Lett. B 522, 315 (2001) [arXiv:hep-ph/0109111].
[5] T. W. Kephart and H. Pas, Phys. Rev. D 70, 086009 (2004) [arXiv:hep-ph/0402228].
[6] P. H. Frampton, R. M. Rohm and T. Takahashi, Phys. Lett. B 570, 67 (2003)
[arXiv:hep-ph/0302074].
http://arxiv.org/abs/hep-th/9803015
http://arxiv.org/abs/hep-th/0603065
http://arxiv.org/abs/hep-th/9802183
http://arxiv.org/abs/hep-ph/0109111
http://arxiv.org/abs/hep-ph/0402228
http://arxiv.org/abs/hep-ph/0302074
[7] P. H. Frampton and T. W. Kephart, Phys. Lett. B 585, 24 (2004) [arXiv:hep-ph/0306053].
[8] P. H. Frampton and T. Takahashi, Phys. Rev. D 70, 083530 (2004) [arXiv:hep-ph/0402119].
[9] P. H. Frampton, Mod. Phys. Lett. A22 (2007, in press), [astro-ph/0607391].
[10] P. H. Frampton, Mod. Phys. Lett. A 22, 347 (2007) [arXiv:hep-ph/0610116].
[11] P. H. Frampton, Phys. Rev. D 60, 085004 (1999) [arXiv:hep-th/9905042].
[12] For a recent detailed phenomological study of trinification see, J. Sayre, S. Wiesenfeldt, and
S. Willenbrock, Phys. Rev. D 73, 035013 (2006) [arXiv:hep-ph/0601040].
[13] P. H. Frampton, Phys. Rev. D 60, 121901 (1999) [arXiv:hep-th/9907051].
[14] S. Martin, Phys. Rev. D 46, 2769 (1992) [arXiv:hep-ph/9207218].
[15] S. Willenbrock, Phys. Lett. B 561, 130 (2003) [arXiv:hep-ph/0302168].
http://arxiv.org/abs/hep-ph/0306053
http://arxiv.org/abs/hep-ph/0402119
http://arxiv.org/abs/astro-ph/0607391
http://arxiv.org/abs/hep-ph/0610116
http://arxiv.org/abs/hep-th/9905042
http://arxiv.org/abs/hep-ph/0601040
http://arxiv.org/abs/hep-th/9907051
http://arxiv.org/abs/hep-ph/9207218
http://arxiv.org/abs/hep-ph/0302168
	Introduction
	 non-SUSY Z7
	 Initial Particle Content and sin2W
	 Symmetry Breaking
	 Low Energy Particle Content 
	Gauged R-parity and Dark Matter Candidates
	Proton Decay
	Conclusions
	Acknowledgments
	References

  Dark matter candidates and proton decay in a class of models based on the
AdS/CFT correspondence are discussed. We show that the present bound on the
proton decay lifetime is inconsistent with ${\cal N} = 1$ SUSY, and strongly
constrains ${\cal N} = 0$ non-SUSY, low scale trinification type unification of
orbifolded AdS$\otimes S^5$ models.

Introduction
	 non-SUSY Z7
	 Initial Particle Content and sin2W
	 Symmetry Breaking
	 Low Energy Particle Content 
	Gauged R-parity and Dark Matter Candidates
	Proton Decay
	Conclusions
	Acknowledgments
	References
