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arXiv:hep-ph/0305116v1 12 May 2003

hep-ph/0305116

KUNS-1843

February 1, 2008

Sliding Singlet Mechanism Revisited

Nobuhiro Maekawaaand Toshifumi Yamashitab

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

Abstract

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.

ae-mail: maekawa@gauge.scphys.kyoto-u.ac.jp

be-mail: yamasita@gauge.scphys.kyoto-u.ac.jp

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1 Introduction

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[7], the GIFT mechanism[8], and via orbifold

boundary condition[9].

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[10], 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[11], whose anomaly is cancelled by the Green-Schwarz

mechanism[12], 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[14] 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[15].

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.

2.1

SU(5)

The sliding singlet mechanism was originally proposed in the context of SU(5)[1], in which

the following terms are allowed in the superpotential;

Wss=¯H(A + Z)H.

(2.1)

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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,

????¯H???2+ |?H?|2?

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[2].

VSUSY= |FH|2+ |F ¯ H|2=

|−3v + ?Z?|2,

(2.2)

? m2v

?H?2+? m2∼ MGUTto minimize the potential.

2.2

SU(6)

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

SBM2

GUT(≫ M4

SB) if

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• 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

and unphysical.

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′,

(2.3)

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.[3], four pairs are introduced and the relevant part of the super-

potential is given as

W

= W(A) + W(¯Hi,Hi)

?

i=1,2

+

ai¯ H′i(A + Zi)Hi+

?

i=1,2

¯ ai¯Hi(A +¯Zi)H′

i,

(2.4)

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):

IA

I ¯ H′

1

I ¯ H′

¯IA

MI

¯ a1

¯ a2

¯IH′

1

a1?H1?

¯IH′

2

a2?H2?

¯IH1

02αI¯ a1v

¯IH2

002αI¯ a2v

MI=

2

I ¯ H1

0

I ¯ H2

0

0

?¯H1

0

0

??¯H2

0

0

0

?

2αIa1v

0

c|?H2?|2

−c?H∗

2αIa2v

−c?H1H∗

c|?H1?|2

2?

1H2?

.

(2.5)

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.[4] 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

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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

previous model.3

The author of Ref.[4] 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

general case.

3 Generalization

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

breaking effects.

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.[3]. 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′

iH′

i, are needed.

4