# Singlet Extensions of the MSSM withZ4R Symmetry

Authors: Michael Ratz; Patrick K. S. Vaudrevange
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

We discuss singlet extensions of the MSSM withZ4R symmetry. We show that
holomorphic zeros can avoid a potentially large coefficient of the term linear
in the singlet. The emerging model has both an effective μ term and a
supersymmetric mass term for the singlet μN which are controlled
by the gravitino mass. The μ term turns out to be suppressed against μN by about one or two orders of magnitude. We argue that this
class of models might provide us with a solution to the little hierarchy
problem of the MSSM.


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## Body

## 1. Purpose of This Paper

TheZ4R symmetry [1, 2] provides us with compelling solutions of the μ and proton decay problems of the minimal supersymmetric extension of the standard model (MSSM). This symmetry appears anomalous, but the anomaly is cancelled by the (discrete) Green-Schwarz (GS) mechanism [3] in such a way that it does not spoil gauge coupling unification (see, e.g., [4] for a discussion). More precisely, if one extends the MSSM by a symmetry (continuous or discrete) that solves the μ problem and (i) demands anomaly freedom (while allowing GS anomaly cancellation), (ii) demands that the usual Yukawa couplings and the Weinberg operator be allowed, (iii) demands consistency with SO(10) grand unification, and (iv) demands precision gauge coupling unification, then this Z4R is the unique solution [2] (see also [5] for an alternative proof). By relaxing (iii) to consistency with SU(5), one obtains four additional symmetries [6]. Further, Z4R can be thought of as a discrete remnant of the Lorentz symmetry of compact extra dimensions; that is, it has a simple geometric interpretation and can arise in explicit string-derived models with the precise MSSM matter content [7]. The charge assignment is very simple: MSSM matter superfields have Z4R charge 1 while the Higgs superfields have 0, and the superpotential W carries R charge 2.However, if one attempts to construct singlet extensions of theZ4R MSSM, one faces the problem that the presence of superpotential coupling of the singlet N to the Higgs bilinear HuHd implies that also a linear term in the singlet is allowed by all symmetries. In more detail, since the Higgs bilinear has Z4R charge 0, the singlet N needs to carry charge 2 in order to match the Z4R charge 2 of the superpotential. Then the desired term W⊂NHuHd is allowed. However, in this case one might expect to have a problematic, unsuppressed linear term in N in the (effective) superpotential, (1)Weff⊃Λ2N,with Λ of the order of the fundamental scale. In order to forbid this linear term, one may try to add a new symmetry. It is quite straightforward to see that an ordinary symmetry cannot forbid this linear term and be, at the same time, consistent with criteria (i)–(iv) above: in order to forbid the linear term, the singlet N needs to carry a nontrivial charge under the new symmetry. But, as we want the term NHuHd, this implies that also HuHd carries a nontrivial charge. Consequently, the new symmetry would yield a solution to the μ problem. However, this is not possible: as stated above, one can prove that (under our assumptions) the unique solution to the μ problem is Z4R, and this symmetry does not forbid the linear term.In this paper, we take an alternative route and describe how one can get rid of the linear term (1) by employing holomorphic zeros [8] associated with an additional pseudoanomalous U(1) gauge symmetry.


## 2. Forbidding the Linear Term in theZ4R (G)NMSSM

### 2.1. Setup

Consider a singlet extension of the MSSM with a singletN and an additional Z4R×U(1)anom symmetry. U(1)anom is pseudoanomalous U(1) symmetry, whose anomaly is cancelled by the GS mechanism. Such U(1) factors often arise in string compactifications and are accompanied by nontrivial Fayet-Iliopoulos (FI) term [9] ξ, which arises at 1-loop [10]. The FI term of the U(1)anom is assumed to be cancelled by a nontrivial vacuum expectation value (VEV) of a “flavon” ϕ, which carries negative U(1)anom charge and Z4R charge 0. Without loss of generality, we can normalize U(1)anom such that ϕ has charge -1 and ξ>0. (Of course, in true string-derived models the situation is usually more complicated: in approximately 500 out of a total of 11940 MSSM-like models from [11] the FI term can be cancelled with one field only. In all other models, one would have to identify ϕ with an appropriate monomial of MSSM singlet fields (see Appendix A for details).) For the sake of definiteness, we assume that(2)ε≔ϕMP~sin⁡ϑCabibbo~0.2,where the Planck scale MP is identified with the “fundamental scale.” In this case, U(1)anom can be used as Froggatt-Nielsen symmetry [12] to explain the flavor structure of quarks and leptons. However, this assumption is not crucial for the subsequent discussion, yet this is what one gets in explicit orbifold compactifications of the heterotic string which exhibit the exact MSSM spectrum at energies below the compactification scale.Further, also the anomaly ofZ4R is assumed to be cancelled by the GS mechanism with the GS axion being contained in the dilaton or another superfield, which we will denote by S. Since the mixed U(1)anom-GSM2 and Z4R-GSM2 anomalies are universal, the GS mechanism does not interfere with the beautiful picture of MSSM gauge coupling unification (see, e.g., [4]). The “nonperturbative” term e-bS carries the same Z4R charge as the superpotential, namely, 2. It might be thought of as some nonperturbative hidden sector (see, e.g., [13]). Further, e-bS will also carry positive U(1)anom charge s>0 such that holomorphic zeros get lifted by “nonperturbative” terms. More details on the charge of e-bS can be found in Appendix B (see, e.g., [6, 14]). In more detail, we demand that (3)Whid~MP3ϕMPse-bSbe allowed, which is equivalent to the statement that e-bS carries U(1)anom charge s>0. (Note that s may also be fractional even if the charges of all “fundamental” fields are integer, for instance, if one assumes that Whid is given by the Affleck-Dine-Seiberg superpotential [15]. Examples for such terms can be found, e.g., in [13].) Whid may be thought of as gaugino condensate [16] or some other nonperturbative physics, such as the one discussed in [17], which is involved in spontaneous supersymmetry breaking. We discuss this in more detail in Appendix B. Inserting the ϕ VEV we obtain (4)Whid→ϕ→ϕMP2m3/2in Planck units. (Note that (3) is not the “full” hidden sector superpotential. One must, of course, make sure that ϕ does not attain an F-term VEV, and one needs to cancel the vacuum energy. A detailed discussion of these issues is, however, beyond the scope of the present paper.) This implies, in particular, that (5)e-bS~m3/2MPε-s.That is, R symmetry breaking is controlled by the gravitino mass, as it should be, and due to the presence of U(1)anom we obtain a Froggatt-Nielsen-like [12] modification of the terms. However, in contrast to the usual Froggatt-Nielsen mechanism, it yields in our setup an enhancement rather than a suppression factor for the lifting of the holomorphic zeros by nonperturbative effects.


### 2.2. Charges and Allowed Terms in the Superpotential

We summarize theU(1)anom and Z4R charges in Table 1.Table 1
Charge assignment.
ϕ
H
u
H
d
N
e
-
b
S
U
(
1
)
anom
-
1
h
>
0
n
<
0
s
>
0
Z
4
R
0
0
2
2Below theU(1)anom breaking scale set by the ϕ VEV, we wish to have a nontrivial μ term at the nonperturbative level; that is,(6)Weff⊃MPe-bSϕMPs+hHuHd.This implies (7)s+h≥0.We will then get effectively (8)Weff⊃MPe-bSϕMPs+hHuHd≕μHuHdwithμ~m3/2εh.Next, we wish to couple the singlet N to the Higgs bilinear. We hence demand that(9)n+h≥0such that(10)Weff⊃ϕMPn+hNHuHd~εn+hNHuHd≕λNHuHdwithλ~εn+h.Now we wish to forbid the linear term in N at the perturbative level. This can be achieved with holomorphic zeros [8], which amounts in our setup to demanding that(11)n<!0.This implies, in particular, that the cubic term in N is also forbidden.Of course, this all works only if we make sure thatϕ rather than N cancels the FI term. This might be achieved by postulating that the soft mass squared of N is positive while the one of ϕ is negative; that is,(12)m~ϕ2<0,m~N2>0.Full justification of such an assumption would require deriving the setting from some UV complete construction such as a string model. This is, however, beyond the scope of this paper.We further obtain nonperturbative terms which are linear or quadratic inN if n+2s≥0 or 2n+s≥0, respectively. Altogether we have (13a)n+h≥0⟺couplingλbetweenNandHuHdwithλ~εn+h,(13b)n<0⟺suppress  linear  term  inN,(13c)s+h≥0⟺μterm  withμ~MPεs+he-bS~εhm3/2,(13d)n+2s≥0⟺f2Nterm  withf~MPεn+2s/2e-bS~εn/2m3/2,(13e)2n+s≥0⟺μNN2term  withμN~MPε2n+se-bS~ε2nm3/2,(13f)3n+2s≥0⟺κN3term  withκ~ε3n+2se-bS2~ε3nm3/22MP2,where the coefficient κ of the cubic term is generically highly suppressed. Not all conditions on {n,h,s} are independent; for example, if the quadratic term is allowed also, since s>0, the linear term will be present.There are many possible values that satisfy all the constraints; for instance,{n,h,s}={-1,1,2}, which gives us(14)λ~O1,μ~εm3/2,μN~1ε2m3/2,f~1εm3/2.That is, the (holomorphic) μ term is roughly two orders of magnitude smaller than μN, which might be favorable in view of the so-called “little hierarchy problem.”Note also that the effective superpotential(15)Weff=f2N+μHdHu+λNHdHu+μNN2admits two solutions to the F- and D-term equations, the first one being (recall that n<0) (16a)N=-μλ~-εnm3/2,(16b)Hu=Hd=2μμN-λf2λ~ε-h/2m3/2.Here one has electroweak symmetry breaking prior to supersymmetry breaking, and the Higgs VEV may be subject to cancellations since both μμN and λf2 are of the order ε2n+h, for example, ε-1 in our example. The second solution is (17a)N=-f22μN~-12εnm3/2,(17b)Hu=Hd=0with unbroken electroweak symmetry for unbroken supersymmetry.


### 2.3. Discussion

In summary, we find that theZ4R×U(1)anom charge assignment of Table 1 yields an effective superpotential, (18)Weff=f2N+μHdHu+λNHdHu+μNN2,with all the dimensionful parameters μ, μN, and f of the order of the gravitino mass m3/2. This description is valid below the U(1)anom breaking scale, which is set by the flavon VEV ϕ. In particular, the linear term in the singlet is sufficiently suppressed. In contrast to the original (G)NMSSM [18], here,(i)
there is (essentially) no cubic term inN;(ii)
there is a suppressed linear term inN. (Note that, unlike in [18], we cannot shift the singlet in order to eliminate the linear term because the point N=0 is special as it denotes the point of unbroken Z4R.)The scheme leads to certain predictions and expectations: (1)
Forbidding the linear term by holomorphic zeros implies the absence of a perturbative cubic term inN.(2)
Further, we obtain the “little hierarchies” (recall thatn<0) (19)μ~εh+2nμN,f~μεh+n/2.


### 2.4. Further Applications

Clearly, this method of avoiding a linear term in a gauge singlet may find further applications. For instance, in model building one sometimes introduces so-called “driving fields” in order to “explain” a certain structure of flavon VEVs. Here, one may forbid too large tadpole terms in the same way as we have discussed above.


## 2.1. Setup

Consider a singlet extension of the MSSM with a singletN and an additional Z4R×U(1)anom symmetry. U(1)anom is pseudoanomalous U(1) symmetry, whose anomaly is cancelled by the GS mechanism. Such U(1) factors often arise in string compactifications and are accompanied by nontrivial Fayet-Iliopoulos (FI) term [9] ξ, which arises at 1-loop [10]. The FI term of the U(1)anom is assumed to be cancelled by a nontrivial vacuum expectation value (VEV) of a “flavon” ϕ, which carries negative U(1)anom charge and Z4R charge 0. Without loss of generality, we can normalize U(1)anom such that ϕ has charge -1 and ξ>0. (Of course, in true string-derived models the situation is usually more complicated: in approximately 500 out of a total of 11940 MSSM-like models from [11] the FI term can be cancelled with one field only. In all other models, one would have to identify ϕ with an appropriate monomial of MSSM singlet fields (see Appendix A for details).) For the sake of definiteness, we assume that(2)ε≔ϕMP~sin⁡ϑCabibbo~0.2,where the Planck scale MP is identified with the “fundamental scale.” In this case, U(1)anom can be used as Froggatt-Nielsen symmetry [12] to explain the flavor structure of quarks and leptons. However, this assumption is not crucial for the subsequent discussion, yet this is what one gets in explicit orbifold compactifications of the heterotic string which exhibit the exact MSSM spectrum at energies below the compactification scale.Further, also the anomaly ofZ4R is assumed to be cancelled by the GS mechanism with the GS axion being contained in the dilaton or another superfield, which we will denote by S. Since the mixed U(1)anom-GSM2 and Z4R-GSM2 anomalies are universal, the GS mechanism does not interfere with the beautiful picture of MSSM gauge coupling unification (see, e.g., [4]). The “nonperturbative” term e-bS carries the same Z4R charge as the superpotential, namely, 2. It might be thought of as some nonperturbative hidden sector (see, e.g., [13]). Further, e-bS will also carry positive U(1)anom charge s>0 such that holomorphic zeros get lifted by “nonperturbative” terms. More details on the charge of e-bS can be found in Appendix B (see, e.g., [6, 14]). In more detail, we demand that (3)Whid~MP3ϕMPse-bSbe allowed, which is equivalent to the statement that e-bS carries U(1)anom charge s>0. (Note that s may also be fractional even if the charges of all “fundamental” fields are integer, for instance, if one assumes that Whid is given by the Affleck-Dine-Seiberg superpotential [15]. Examples for such terms can be found, e.g., in [13].) Whid may be thought of as gaugino condensate [16] or some other nonperturbative physics, such as the one discussed in [17], which is involved in spontaneous supersymmetry breaking. We discuss this in more detail in Appendix B. Inserting the ϕ VEV we obtain (4)Whid→ϕ→ϕMP2m3/2in Planck units. (Note that (3) is not the “full” hidden sector superpotential. One must, of course, make sure that ϕ does not attain an F-term VEV, and one needs to cancel the vacuum energy. A detailed discussion of these issues is, however, beyond the scope of the present paper.) This implies, in particular, that (5)e-bS~m3/2MPε-s.That is, R symmetry breaking is controlled by the gravitino mass, as it should be, and due to the presence of U(1)anom we obtain a Froggatt-Nielsen-like [12] modification of the terms. However, in contrast to the usual Froggatt-Nielsen mechanism, it yields in our setup an enhancement rather than a suppression factor for the lifting of the holomorphic zeros by nonperturbative effects.


## 2.2. Charges and Allowed Terms in the Superpotential

We summarize theU(1)anom and Z4R charges in Table 1.Table 1
Charge assignment.
ϕ
H
u
H
d
N
e
-
b
S
U
(
1
)
anom
-
1
h
>
0
n
<
0
s
>
0
Z
4
R
0
0
2
2Below theU(1)anom breaking scale set by the ϕ VEV, we wish to have a nontrivial μ term at the nonperturbative level; that is,(6)Weff⊃MPe-bSϕMPs+hHuHd.This implies (7)s+h≥0.We will then get effectively (8)Weff⊃MPe-bSϕMPs+hHuHd≕μHuHdwithμ~m3/2εh.Next, we wish to couple the singlet N to the Higgs bilinear. We hence demand that(9)n+h≥0such that(10)Weff⊃ϕMPn+hNHuHd~εn+hNHuHd≕λNHuHdwithλ~εn+h.Now we wish to forbid the linear term in N at the perturbative level. This can be achieved with holomorphic zeros [8], which amounts in our setup to demanding that(11)n<!0.This implies, in particular, that the cubic term in N is also forbidden.Of course, this all works only if we make sure thatϕ rather than N cancels the FI term. This might be achieved by postulating that the soft mass squared of N is positive while the one of ϕ is negative; that is,(12)m~ϕ2<0,m~N2>0.Full justification of such an assumption would require deriving the setting from some UV complete construction such as a string model. This is, however, beyond the scope of this paper.We further obtain nonperturbative terms which are linear or quadratic inN if n+2s≥0 or 2n+s≥0, respectively. Altogether we have (13a)n+h≥0⟺couplingλbetweenNandHuHdwithλ~εn+h,(13b)n<0⟺suppress  linear  term  inN,(13c)s+h≥0⟺μterm  withμ~MPεs+he-bS~εhm3/2,(13d)n+2s≥0⟺f2Nterm  withf~MPεn+2s/2e-bS~εn/2m3/2,(13e)2n+s≥0⟺μNN2term  withμN~MPε2n+se-bS~ε2nm3/2,(13f)3n+2s≥0⟺κN3term  withκ~ε3n+2se-bS2~ε3nm3/22MP2,where the coefficient κ of the cubic term is generically highly suppressed. Not all conditions on {n,h,s} are independent; for example, if the quadratic term is allowed also, since s>0, the linear term will be present.There are many possible values that satisfy all the constraints; for instance,{n,h,s}={-1,1,2}, which gives us(14)λ~O1,μ~εm3/2,μN~1ε2m3/2,f~1εm3/2.That is, the (holomorphic) μ term is roughly two orders of magnitude smaller than μN, which might be favorable in view of the so-called “little hierarchy problem.”Note also that the effective superpotential(15)Weff=f2N+μHdHu+λNHdHu+μNN2admits two solutions to the F- and D-term equations, the first one being (recall that n<0) (16a)N=-μλ~-εnm3/2,(16b)Hu=Hd=2μμN-λf2λ~ε-h/2m3/2.Here one has electroweak symmetry breaking prior to supersymmetry breaking, and the Higgs VEV may be subject to cancellations since both μμN and λf2 are of the order ε2n+h, for example, ε-1 in our example. The second solution is (17a)N=-f22μN~-12εnm3/2,(17b)Hu=Hd=0with unbroken electroweak symmetry for unbroken supersymmetry.


## 2.3. Discussion

In summary, we find that theZ4R×U(1)anom charge assignment of Table 1 yields an effective superpotential, (18)Weff=f2N+μHdHu+λNHdHu+μNN2,with all the dimensionful parameters μ, μN, and f of the order of the gravitino mass m3/2. This description is valid below the U(1)anom breaking scale, which is set by the flavon VEV ϕ. In particular, the linear term in the singlet is sufficiently suppressed. In contrast to the original (G)NMSSM [18], here,(i)
there is (essentially) no cubic term inN;(ii)
there is a suppressed linear term inN. (Note that, unlike in [18], we cannot shift the singlet in order to eliminate the linear term because the point N=0 is special as it denotes the point of unbroken Z4R.)The scheme leads to certain predictions and expectations: (1)
Forbidding the linear term by holomorphic zeros implies the absence of a perturbative cubic term inN.(2)
Further, we obtain the “little hierarchies” (recall thatn<0) (19)μ~εh+2nμN,f~μεh+n/2.


## 2.4. Further Applications

Clearly, this method of avoiding a linear term in a gauge singlet may find further applications. For instance, in model building one sometimes introduces so-called “driving fields” in order to “explain” a certain structure of flavon VEVs. Here, one may forbid too large tadpole terms in the same way as we have discussed above.


## 3. Discussion

We have discussed how to build singlet extensions of the MSSM withZ4R symmetry. We have shown that a potentially large linear term in the singlet can be avoided by using holomorphic zeros. The resulting model has a μ term, a supersymmetric mass of the order of the gravitino mass m3/2, as well as a coefficient of an effective linear term in the singlet of the order m3/22. μ is expected to be one or two orders of magnitude smaller than μN. This might be viewed as the first step towards a solution to the little hierarchy problem; that is, explain why the electroweak scale is at least one order of magnitude smaller than the soft supersymmetric terms. Obtaining a complete solution requires the derivation of our setting from a UV complete model, which allows us to compute various terms precisely. This, however, is beyond the scope of this paper.



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*Source: 785217-2015-10-01.xml*