arXiv:hep-ph/0305116v1 12 May 2003
February 1, 2008
Sliding Singlet Mechanism Revisited
Nobuhiro Maekawaaand Toshifumi Yamashitab
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
We show that the unification of the doublet Higgs in the standard model (SM)
and the Higgs to break the grand unified theory (GUT) group stabilizes the sliding
singlet mechanism which can solve the doublet-triplet (DT) splitting problem. And
we generalize this attractive mechanism to apply it to many unified scenarios. In this
paper, we try to build various concrete E6unified models by using the generalized
sliding singlet mechanism.
The well-known success of the gauge coupling unification in the minimal supersymmetric
standard model (MSSM) likely supports the attractive idea of supersymmetric grand
unified theory (SUSY-GUT). On the other hand, we know there are some obstacles in
constructing a realistic SUSY-GUT. One of the biggest problems is the so-called DT
splitting problem. Generically in SUSY-GUTs, there are color triplet partners of the
MSSM Higgs, and the nucleon decay via dimension five operators becomes too rapid.
In order to suppress this proton decay, the color triplet partners must have very large
mass (≫ MGUT ∼ 1016GeV), in contrast to the doublet Higgs whose mass has to be
of order the weak scale MW ∼ 102GeV. Some ideas to solve this problem have been
proposed : the sliding-singlet mechanism[1, 2, 3, 4], the missing partner mechanism[5, 6],
the Dimopoulos-Wilczek (DW) mechanism, the GIFT mechanism, and via orbifold
Among these ideas, the first mechanism is the smartest solution which realizes the
DT splitting dynamically. Although it was shown that the originally proposed SU(5)
model cannot act effectively if SUSY breaking effect is considered, some authors have
proposed SU(6) extensions in which this mechanism acts without destabilization due to
SUSY breaking[2, 3, 4]. In this paper, we abstract the essence of this sliding singlet
mechanism in SU(6) models and generalize it to apply to many other unified theories.
Actually in E6unification it is found that for many directions of VEV of the adjoint Higgs
this mechanism may act. Corresponding to these breaking patterns, we construct some
E6Higgs sectors in which the DT splitting problem is indeed solved through this mecha-
nism. Several concrete models are propose in the context of the SUSY-GUT in which an
anomalous U(1)Agauge symmetry, whose anomaly is cancelled by the Green-Schwarz
mechanism, plays an important role[13, 14, 15, 16, 17, 18, 20] in solving various prob-
lems in SUSY scenario. And we examine whether the already proposed realistic quark and
lepton sector is compatible with such a Higgs sector or not. Note that this E6group is
interesting as a unified group, in the sense that the SUSY flavor problem can be solved in
E6SUSY-GUT with anomalous U(1)Aand non-abelian horizontal gauge symmetry.
In section 2, we briefly review the sliding singlet mechanism in the context of SU(5)
and SU(6). In section 3, we generalize this mechanism to the general gauge group. In
section 4, we construct some concrete Higgs sectors.
2 The Sliding Singlet Mechanism
In this section, We review the present status of the sliding singlet mechanism. For this
purpose, we sometimes omit details, which are described in each references.
The sliding singlet mechanism was originally proposed in the context of SU(5), in which
the following terms are allowed in the superpotential;
Wss=¯H(A + Z)H.
Here, the adjoint Higgs A(24) is assumed to have the VEV ?A? = diag(2,2,2,−3,−3)v
which breaks SU(5) into SU(3)C×SU(2)L×U(1)Y (GSM), and the (anti)fundamental
Higgs H(5) and¯H(¯5) contain the MSSM doublet Higgs, Huand Hd, respectively. Since
the doublet Higgs have non-vanishing VEVs ?Hu? and ?Hd? to break SU(2)L×U(1)Yinto
U(1)EM, the minimization of the potential,
leads to the vanishing doublet Higgs mass µ = (?A? + ?Z?)2= −3v + ?Z? = 0 by sliding
the VEV of the singlet Higgs Z(1)1. For these VEVs, ?A? +?Z? = diag(5,5,5,0,0)v, the
color triplet partners of doublet Higgs have a large mass 5v ∼ 1016GeV.
Unfortunately, it is known that if SUSY breaking is taken into account, this DT
splitting is failed. For example, the soft SUSY breaking mass term ? m2|Z|2(? m ∼ MSB)
shifts the VEV ?Z? by an amount of δ?Z? ∼
Thus the doublet-triplet splitting is spoiled by SUSY breaking effect in this mechanism.
2This is caused by the fact that the terms |FH|2+|F ¯ H|2give only a mass of order ?H? to
Z, which are the same order as (or smaller than) the SUSY breaking contribution. Since
this mass parameterizes the stability of ?Z? against other contributions to the potential,
e.g. SUSY breaking effects ? m2|Z|2, soft terms of order MSB easily shift the VEV from
that in the SUSY limit by a large amount.
It is obvious that this problem can be solved if we take the large VEVs ?H? and?¯H?
(larger than√MSBMGUT) which give larger mass to Z and stabilize the VEV of Z against
SUSY breaking effects. Of course, it is not consistent with the experiments to take such
large VEVs for SM doublet Higgs. But for other Higgs, for example, that breaks a larger
gauge group into the SM gauge group, we can take such large VEV. This is an essential
idea of Sen.
VSUSY= |FH|2+ |F ¯ H|2=
|−3v + ?Z?|2,
?H?2+? m2∼ MGUTto minimize the potential.
SU(6) is the simplest candidate for the above purpose, and some authors examine the
possibility[2, 3]. The relevant part of the superpotential is written in a similar form as
(2.1), where A(35) has a VEV ?A? = diag(1,1,1,−1,−1,−1)v which breaks SU(6) into
SU(3)C×SU(3)L×U(1) and H(6) (¯H(¯6)) denotes (anti)fundamental Higgs which has a
VEV in SU(5) singlet component. Note that the simplest extension, in which one pair of
(anti)fundamental Higgs is introduced, cannot act effectively. This is because
• the F-term of Z gives a contribution of O(M4
1Here, the contributions to the potential from FA and FZ are neglected, because they are of order
(?¯HH?)2. In this sense, the doublet Higgs mass µ does not vanish exactly but may become of order
SUSY breaking scale MSB.
2The soft term ? mZFZ also destabilizes the sliding singlet mechanism because this term alters the
contribution of FZ to the scalar potential as
?H? is order of the weak scale. Such a term is induced by loop effects through the coupling between Z
and the color triplet Higgs. Therefore, even if the terms ? mZFZ and ? m2|Z|2are absent at the tree level,
this problem cannot be avoided.
GUT) to the scalar potential.
?? ¯HH + ? mZ??2, that is the order of M2
• the term¯HAH gives a contribution to the F-flatness condition of A which destabi-
lizes the required form of VEV of A, if ?A? is determined from FA.
• this term also gives mass terms of 5 and¯5 of SU(5),?¯H?AH and¯HA?H?.
• one pair of doublet of (anti)fundamental Higgs is the Nambu-Goldstone (NG) mode
The first two and the last one are resolved if one of the pair of (anti)fundamental Higgs
has vanishing VEV. Thus, (2.1) is altered as
Wss=¯ H′(A + Z)H +¯H(A + Z)H′,
where primed fields have vanishing VEVs. Because of the third reason, at least one more
pair of (anti)fundamental Higgs is required.
For example in Ref., four pairs are introduced and the relevant part of the super-
potential is given as
= W(A) + W(¯Hi,Hi)
ai¯ H′i(A + Zi)Hi+
¯ ai¯Hi(A +¯Zi)H′
where aiand ¯ aiare coupling constants and W(A) and W(¯Hi,Hi) are some sets of terms
which give the desired VEV (as one of discrete vacua) to A and¯Hi,Hirespectively. This
gives following mass matrix of 5 ×¯5 of SU(5):
I ¯ H′
I ¯ H′
I ¯ H1
I ¯ H2
Here α2= 0 and α3= 1, which are realized by the sliding singlet mechanism, and M3= 0
because 3Aand¯3Aare NG modes by breaking SU(6) → SU(3)C× SU(3)L× U(1). M2
and c are determined from W(A) and W(¯Hi,Hi), respectively. From this mass matrix, it
can be found that there are two massless modes for I = 2 and one for I = 3. Since one
pair of 5 and¯5 of SU(5) are absorbed by the Higgs mechanism, only one pair of doublets
remains massless and therefore the DT splitting is realized. In this model, the massless
modes come from a linear combination of the primed fields.
Note that, due to the large VEVs of Hiand¯Hi, this hierarchy is stable against the
SUSY breaking corrections, which means that all the elements of the mass matrix have
corrections due to SUSY breaking at most O(MSB).
Another example was proposed in Ref. in the context of SU(6) × SU(2). The
relevant part is similar as previous model except the absence of the indices i of the
singlet Higgs. The indices of the (anti)fundamental Higgs are understood as those of the
symmetry SU(2). This symmetry guarantees that the doublet components of H2 and
¯H2 are massless even if they do not have non-vanishing VEVs. This means that the
doublets are not NG modes and therefore physical modes. In this model, the SM doublet
Higgs comes from the unprimed fields which have non-vanishing VEVs, in contrast to the
The author of Ref. mentions that this is because the symmetry SU(2) relates the
doublets to NG modes in H1 and¯H1. To be more precise, the doublets and the NG
modes belong to a single multiplet of the SU(2) symmetry to which the mass parameter
(?A?+?Z?) respect. However, in the spirit of the sliding singlet, it may be more appropriate
to say that the mass parameter (?A? + ?Z?) gives the same value for the doublets as for
the SU(5) singlet components of H1and¯H1, which must vanish due to the non-vanishing
VEVs. This observation makes it possible to apply the sliding singlet mechanism in more
Now, we examine how we can generalize the sliding singlet mechanism.
The essential idea of the sliding singlet mechanism is following.
• If the mass parameter of a certain component is guaranteed to be the same value as
that of the other component which has a non-vanishing VEV and the later vanishes
dynamically due to the VEV, the former also vanishes. And if the non-vanishing
VEVs are sufficiently large, the mass hierarchy is stable against possible SUSY
It is the case that a doublet component and a singlet component with non-vanishing
VEV belong to a single multiplet of the symmetry to which the mass terms respect, e.g.
SU(3)C×SU(3)L×U(1) for the previous SU(6) example in Ref.. In this case, the DT
splitting problem can be solved.
Moreover, if the mass parameter depends only on the VEVs of adjoint Higgs and
singlet Higgs, the above condition for the sliding singlet mechanism to act can be easily
examined. This is because the mass parameter for each component is determined by
each quantum number of U(1) which are fixed by the non-vanishing VEV of the adjoint
Higgs. Therefore, if the charge of the doublet Higgs component is the same as that of the
singlet component which has non-vanishing VEV, then the massless doublet Higgs can be
realized by the sliding singlet mechanism. This perspective holds for any gauge group,
even if the mass term involves non-renormalizable terms.
It is obviously important to know the charges for the SM singlet and the doublet Higgs
under the U(1) generator which are fixed by the non-vanishing VEV of the adjoint Higgs.
If a GUT group G includes SU(3)C×SU(2)L×U(1)Y×U(1)r−4as a subgroup, where r is
the rank of G, the U(1) generator must be a linear combination of r −3 U(1) generators.
Therefore, it is helpful to know the charges of these U(1) generators in order to classify
3In order to give masses to the primed fields, some additional terms, e.g.¯H′
i, are needed.