UCBPTH02/11
LBNL49936
Symmetry and the Problem
Lawrence J. Hall, Yasunori Nomura and Aaron Pierce
Department of Physics, University of California, Berkeley, CA 94720, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
1. The possibility that nature becomes supersymmetric at the weak scale offers a well motivated and exciting scenario for physics beyond the standard model. It is well motivated because it allows a dynamical generation of the weak scale, and an understanding of why this scale is much less than the Planck scale. Furthermore, it gives rise to a highly successful numerical prediction for gauge coupling unification. It is exciting because weak scale supersymmetry will be thoroughly tested at hadron colliders over the next decade.
Nevertheless, the underlying structure of the fields and interactions of the weak scale supersymmetric theory contains three puzzles:

In nonsupersymmetric field theories there are three distinct types of fields, corresponding to particles with spin , and . In this case, there is no doubt as to what distinguishes the Higgs field from the lepton doublet field. In contrast, in supersymmetric field theories there are just two types of fields: vector multiplets and chiral multiplets. Hence it is now no longer clear what distinguishes a Higgs field, , from a matter field, . In particular, the down type Higgs and lepton doublets have identical gauge and spacetime properties; what distinguishes the Higgs boson from the sneutrino?

Phenomenologically, interactions cannot be the most general allowed by known gauge and spacetime symmetries. The superpotential must contain interactions of the form for quark and lepton masses, and most probably at the weak scale for neutrino masses. Yet other forms of superpotential interactions, such as , , and are highly constrained by neutrino masses and proton decay. Some of these interactions are either highly suppressed or forbidden.

A superpotential interaction of the form is a special and intriguing case. On one hand it must be highly suppressed since a coefficient of order the Planck scale or unified mass scale would remove the Higgs doublets from the low energy theory. On the other hand it cannot vanish since otherwise there is a massless charged fermion coming from the Higgs/vector multiplets. Indeed the theory is only realistic if the coefficient, , is of order the weak scale, leading to the well known problem. Why should this supersymmetric parameter be of order the supersymmetry breaking scale?
The other great mystery of low energy supersymmetry is the origin of supersymmetry breaking. Like the supersymmetric interactions, a great deal about the structure of the supersymmetry breaking interactions is governed by the requirement of consistency with experiment. However, nothing determines the “messenger” scale, : the highest scale at which the supersymmetry breaking interactions of squarks, sleptons, Higgs and gauginos are local. In supergravity theories this locality is maintained up to the Planck scale [1], but other methods of transporting supersymmetry breaking to the standard model superpartners, such as gauge mediation [2, 3], gaugino mediation [4], and boundary condition supersymmetry breaking [5], have messenger scales that can be anywhere between the weak and Planck scales.
In this paper we study the consequences of imposing an symmetry on the supersymmetric interactions of effective theories with weak scale supersymmetry. This symmetry will allow us to address all three of the puzzles listed above in the the context of supersymmetry breaking with an arbitrary messenger scale. We will describe a continuous symmetry, , although a discrete subgroup is sufficient for our purposes. The soft supersymmetry breaking operators break the symmetry, since they include Majorana gaugino masses, and therefore holomorphic, term, scalar interactions. Using this symmetry, we find a new mechanism for solving the problem for arbitrary messenger scales, and this requires a unique choice for the quantum numbers of matter and Higgs fields. Furthermore, we find that this is also the unique choice which accounts for the absence of , and superpotential interactions, while allowing consistency with quarklepton unification. Finally, this symmetry forces a distinction between Higgs and lepton doublets.
There are two wellknown classes of solutions to the problem. One class corresponds to modifying physics at the Planck scale by adding nonrenormalizable operators [6]; the other changes the physics at the weak scale, as is the case in the NexttoMinimal Supersymmetric Standard Model (NMSSM) [7]. The first approach requires supersymmetry breaking to be mediated at the Planck scale, while the second requires a departure in the weak scale theory from the Minimal Supersymmetric Standard Model (MSSM). Neither of these requirements apply to our mechanism.
In our mechanism, we suppose that all supersymmetric interactions respect some global symmetry, , which forbids and commutes with both gauge and flavor symmetries. To allow for the the possibility of unification of the matter, we require the charges of all matter multiplets to be equal. We consider that some field acquires a vacuum expectation value (vev) at scale between the weak and Planck scales. Supersymmetry breaking interactions, mediated at any scale above , break and cause a deformation in the vacuum resulting in the generation of of order the supersymmetry breaking scale. Such theories have been constructed for the case of mediation at the Planck scale [8]. However, for such high mediation scales is not a problem since the GiudiceMasiero mechanism is available. We study the case of an arbitrary mediation scale. Related solutions for the problem in the context of gauge mediation have been proposed [9], but differ from our mechanism in the origin of the vacuum deformation.
2. If we require that our solution to the problem apply to arbitrary mediation scales, we find that the operator giving rise to is uniquely determined. We assume this operator must be present at tree level — if arises radiatively then the soft term is typically generated at too high a level. Furthermore, must arise from a renormalizable operator. If the operator had a coefficient suppressed by powers of the Planck scale, it would not be possible to get a parameter of the desired size for arbitrary values of the messenger scale. Thus the operator generating is unique:
(1) 
where is a standard model gauge singlet chiral superfield. Our mechanism requires that the supersymmetric interactions break some symmetry at a scale , giving a mass to and forcing . Here and represent the lowest and highest components of the chiral superfield , respectively. Nonzero values for and are generated by supersymmetry breaking.
We discover that our global symmetry, , must be an symmetry from the following argument. The order of magnitude of and generated after supersymmetry breaking can be understood from the charges of both and the soft supersymmetry breaking operators. If is a non symmetry then and have the same transformation, so that both and are generated at the same order in supersymmetry breaking. Hence is of order and much too large: our mechanism requires to be an symmetry.
Now, under the symmetry, there are only two types of supersymmetry breaking terms appearing in the scalar potential. There are holomorphic terms, which we denote by and assign charge , and there are nonholomorphic terms, denoted , which have charge zero. A successful solution to the problem requires , and therefore , to be linear in supersymmetry breaking. This requires that has charge or , so that can be generated proportional to or . Interestingly, this automatically leads to of exactly the right order, since now has charge or and is naturally generated at second order in supersymmetry breaking proportional to or .
Let us consider our two possible charge assignments separately. If has charge , then has total charge . Due to the presumed existence of unification, all matter fields carry the same charge. The Yukawa couplings then force equal quantum numbers for the two Higgs doublets, , so that and . However, in this case the superpotential interaction is allowed, which will push some MSSM matter fields to have masses of order the Planck scale. Therefore, we prefer to consider the charge assignment . Again, the Yukawa couplings require =, so we have
(2) 
This symmetry is extremely powerful: as well as distinguishing between the lepton and Higgs doublet and forbidding , it also forbids and , leading to baryon and lepton number conservation from operators of dimension four and five. Therefore, our solution to the problem has forced us to forbid dangerous dimension four and five operators that might lead to too rapid proton decay. Moreover, if the soft supersymmetry breaking operators provide the only source of breaking, then parity remains unbroken, leading to stability of the lightest superpartner.^{1}^{1}1 parity forbids the generation of dimension four baryon and lepton number violating operators, even after supersymmetry is broken. While dimension five proton decay operators could be generated with small coefficients proportional to supersymmetry breaking, they are phenomenologically irrelevant.
With a mild assumption about the origin of neutrino masses, a completely independent argument will lead us to an identical conclusion for the global symmetry . The Yukawa interactions, , possess a non PecceiQuinn symmetry (PQ: ) as well as the symmetry of Eq. (2). In fact, must be a linear combination of and PQ, or one of its subgroups. The existence of small neutrino masses strongly suggests that the superpotential also contains . Provided that this interaction is not generated by supersymmetry breaking [10], this immediately implies that is neutral under , and hence must be the symmetry given in Eq. (2).
3. The simplest model which realizes the above general mechanism for generating the term is given by the superpotential
(3) 
where and . Here we imagine that the scale is much larger than the weak scale, although this is not necessary for our mechanism to work. We also impose a discrete symmetry , so that the gauge hierarchy is not destabilized by the generation of a large tadpole operator for a singlet field.^{2}^{2}2 We assume that violations of global symmetries by nonperturbative gravitational effects [11] are sufficiently suppressed. Without supersymmetry breaking, the minimum of the potential lies at and , satisfying . There is no flat direction at this level, and all the fields have masses of order .
When we add supersymmetry breaking terms, with a scale of order the weak scale, the vevs will shift. The most general soft supersymmetry breaking terms are given by
(4) 
Here, and , and we have used and to denote the scalar fields of the respective chiral superfields. By minimizing the scalar potential, we obtain and . The vevs of and are both of the order of the weak scale as indicated by the previous general analysis. Therefore, if we introduce couplings to the Higgs doublet and , and terms of order and are generated as
(5) 
(6) 
where and are defined by and . Although for , it is natural to expect since the two parameters run differently under renormalization group evolution; a simple realistic example will be given later. An important point here is that there exists a parameter region where the additional Higgs coupling in Eq. (1) does not change the gross dynamics at the high scale. For instance, if and are sufficiently large, the vev of order is still entirely contained in the field, and the vevs for the Higgs doublets stay smaller than the weak scale. Note that here the various supersymmetry breaking parameters are evaluated at the scale . Thus, both and can be positive without conflicting with electroweak symmetry breaking. Below the scale , the heavy fields and are integrated out and the effective theory contains only the Higgs doublets with the and parameters given by Eqs. (5, 6). The soft supersymmetry breaking parameters must be further evolved from to using the renormalization group equations of this effective theory (MSSM), to evaluate electroweak symmetry breaking.
An important requirement for our mechanism is that must be smaller than the messenger scale of supersymmetry breaking, . Therefore, the superpotential Eq. (3) itself is not sufficient for a complete solution of the problem (except for the supergravity mediation case), since we have introduced by hand a mass parameter, , smaller than the fundamental scale. A complete solution, however, is obtained if we generate the scale by the dynamics of strong gauge interactions. Consider, for example, the gauge theory with four doublet chiral superfields () with the following superpotential:
(7) 
This superpotential explicitly breaks a flavor symmetry of the down to ; and () denote singlet and fivedimensional representations of given by suitable combinations of gauge invariants . The strong dynamics of the gauge theory is described by the effective superpotential
(8) 
where is an additional Lagrange multiplier chiral superfield [12]. For a relatively large value of the coupling , the vacuum lies at and , so that the superpotential is effectively reduced to Eq. (3). Note that the original treelevel superpotential, Eq. (7), does not contain any mass parameters and is invariant under the symmetry with and . In fact, it is the most general superpotential consistent with the combined and symmetries. (A linear term in is forbidden either by requiring that the superpotential not contain any mass parameters, or by imposing an anomalous discrete symmetry under which all the fields are transformed by .) It is also important that does not have an anomaly for (i.e. does not carry charge), so that the previous general argument is not affected by the strong gauge dynamics.
We now consider an application of our mechanism to realistic theories. We find that the mechanism fits beautifully into the framework where small neutrino masses are generated by integrating out righthanded neutrino fields through the seesaw mechanism [13]. We consider the following theory. In addition to the usual three generations of standardmodel quark and lepton superfields, , , , and , we introduce three righthanded neutrino superfields . Here, we have omitted generation indices. The Yukawa couplings are given by
(9) 
We also introduce the gauge symmetry, contained in , under which various fields transform as , , , , and . This gauge symmetry is broken by the vevs of the fields and through the superpotential
(10) 
Here and are gauge invariants consisting of , the doublets under the strong gauge interaction (see discussion around Eqs. (7, 8)). Note that the above superpotentials, Eqs.(9, 10), do not contain any mass parameters and are invariant under the symmetry, , and . For a relatively large , the dynamics of the gauge interaction cause the condensation of , which is transmitted to the vevs of the and fields, . Here, the equality is forced by the term condition. The vevs for all the other fields are zero at this stage: . After introducing soft supersymmetry breaking operators, the vevs of the fields shift. In particular, nonvanishing vevs for and are generated as and , as long as holomorphic soft supersymmetry breaking parameters are not subject to the special relation at the scale . In fact, it is quite natural to expect that the terms for and are different since they are renormalized differently above the scale ; for example, they receive contributions from and gauginos, respectively. Therefore, by introducing the couplings
(11) 
the Majorana masses for the righthanded neutrinos of order and and parameters of order are generated. As in the previous example, the superpotential is the most general renormalizable superpotential consistent with the gauge and global symmetries of the theory, after removing a linear term in as before.
We finally discuss how our general mechanism works explicitly in various supersymmetry breaking scenarios. In gauge mediated supersymmetry breaking, our mechanism requires that the mass of the messenger fields, , is larger than . Since the holomorphic supersymmetry breaking terms ( terms) required for the term generation are small at the messenger scale , they must be generated by renormalization group evolution from to . This can be accomplished, for example, by giving nontrivial or quantum numbers to the messenger fields. In the case of gaugino mediation and boundary condition supersymmetry breaking, our mechanism requires that the compactification scale is larger than . In these cases, the relevant terms of order the weak scale may already exist at the compactification scale, so we do not necessarily have to rely on renormalization group evolution for their generation.^{3}^{3}3 In anomaly mediation [14], the holomorphic supersymmetry breaking parameter associated with the scale is large — of the order of the gravitino mass . Thus is needed to generate a parameter of the correct size. Then, to avoid a too large term, a cancellation between two contributions, such as and in Eq. (6), is required at the level. In any of these mediation mechanisms, is real in the basis where the gaugino masses are real (except for the case of gaugino mediation with treelevel terms), and our origin for and then leads to a real parameter: the supersymmetric problem is solved.
4. In this paper we have proposed an origin for the parameters and of the minimal supersymmetric standard model, which is applicable for any messenger scale, . Although quite general, it does require specific symmetries and interactions. Both and parameters arise from the superpotential interaction . A stage of symmetry breaking occurs at some scale , giving a mass of order to , while determining . Providing the form for the superpotential is guaranteed by an symmetry, with the quantum numbers of Eq. (2), the soft supersymmetry breaking operators, with coefficients and , lead to a small readjustment of the vacuum, giving
(12) 
This symmetry provides a distinction between Higgs and matter superfields, and forbids superpotential interactions that would otherwise lead to baryon number violation at too rapid a rate. Although is broken by supersymmetry breaking, the discrete parity survives so that the lightest superpartner is stable. In the case that the original symmetry is continuous, an axion will be produced by the underlying dynamics which breaks supersymmetry. If all the breaking effects are generated spontaneously (including the constant term in the superpotential needed to cancel the cosmological constant), the dominant mass contribution to the axion will come from the QCD anomaly of the symmetry. In this case, the axion provides a solution to the strong problem [15].
At first sight our symmetry appears to be in conflict with grand unification: since forbids , it also forbids the corresponding mass term for the colored Higgs triplets of unified theories. However, this turns out to be a virtue — such mass terms need to be forbidden to avoid too large a proton decay rate mediated by triplet Higgsino exchange. The colored partners of must become heavy by acquiring mass terms coupling them to other colored states of the theory. This occurs in the missing partner [16] and DimopoulosWilczek [17] mechanisms; however, although these mechanisms are consistent with an underlying symmetry, in the simplest such models is broken at the unification scale, so our generation mechanism may not work in these cases. In contrast, in KaluzaKlein grand unification [18] the desired colored Higgs mass terms arise while preserving symmetry, so that our generation mechanism works well in this case.
The symmetry is so crucial in providing an understanding of the form for the interactions in the superpotential, it is important to seek its origin. Higher dimensional theories are particularly interesting since they have an enlarged set of supersymmetry transformations, which results in a global symmetry in the equivalent four dimensional description. In the case of a five dimensional grand unified theory, compactification breaks the unified gauge symmetry and also the symmetry to , so that precisely the charges considered here may arise [18].
Acknowledgements
Y.N. thanks the Miller Institute for Basic Research in Science for financial support. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the U.S. Department of Energy under Contract DEAC0376SF00098, and in part by the National Science Foundation under grant PHY0098840.
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