Higgs boson from the meta-stable SUSY breaking sector
We construct a calculable model of electroweak symmetry breaking in which the Higgs doublet emerges from the meta-stable SUSY breaking sector as a pseudo Nambu-Goldstone boson. The Higgs boson mass is further protected by the little Higgs mechanism, and naturally suppressed by a two-loop factor from the SUSY breaking scale of 10 TeV. Gaugino and sfermion masses arise from standard gauge mediation, but the Higgsino obtains a tree-level mass at the SUSY breaking scale. At 1 TeV, aside from new gauge bosons and fermions similar to other little Higgs models and their superpartners, our model predicts additional electroweak triplets and doublets from the SUSY breaking sector.
Dynamical supersymmetry (SUSY) breaking has always been difficult due to the existence of nonzero Witten index of supersymmetric pure gauge theories Witten:1982df (). It was believed before that gauge theories breaking SUSY dynamically must either be chiral or have massless matter Dynamical (). Models built along this line turn out to be rather baroque. The situation is greatly improved by the recent discovery of long-lived metastable SUSY breaking vacua in supersymmetric QCD theory by Intriligator, Seiberg and Shih (ISS) Intriligator:2006dd (). The ISS model opens up new possibilities to build realistic models. Using ISS’s idea, a number of new and simple models have been constructed to communicate SUSY breaking (directly) to the standard model Directe Gauge Mediation () R symmetry breaking ().
In an interesting attempt, the ISS model is embedded into the supersymmetric standard model by extending the Higgs sector Abel and Khoze (). But in that work, a strong coupling in the Higgs sector is needed to generate sufficiently large gluino masses. In this paper we will construct an electroweak symmetry breaking model with a weakly coupled Higgs sector embedded inside the meta-stable SUSY breaking sector.
In our model, the soft masses of gauginos and sferminos are generated by the standard gauge mediation. To obtain the soft masses above a few hundred GeV, the lowest allowed SUSY breaking scale is of order 10 TeV. We take two identical ISS sectors with massive fundamental flavors and set the SUSY breaking and the global breaking scale to be of order 10 TeV. We then gauge the diagonal subgroup of the unbroken global symmetry . By adding Yukawa couplings with extra vector-like fermions, both of the symmetries are spontaneously broken at the scale . At the same time the gauge symmetry is broken to the electroweak symmetry. Among the resulting Goldstone boson fields, one doublet is eaten by the heavy gauge bosons, while the other doublet is a pseudo Nambu-Goldstone boson (PNGB) and identified as the Higgs doublet (this Higgs doublet is a light linear combination of the two doublets and in the minimal supersymmetric standard model). The one-loop effective potential of the Higgs doublet also breaks electroweak symmetry at and gives a light Higgs boson with a mass from 100 GeV to 200 GeV. In short, all the mass scales arise from one single scale in our model. Since the scale can be generated dynamically mu generating (), the hierarchy between the electroweak scale and the Planck scale may also be explained.
Below the SUSY breaking scale, the “collective symmetry breaking mechanism” protects the Higgs boson mass as in the little Higgs models little higgs (). Then in our model, like in the super-little Higgs models Roy:2005hg () super little (), the Higgs boson mass is doubly protected by both SUSY and the little Higgs mechanism, and the fine-tuning problem has been alleviated compared to the minimal supersymmetric standard model. However, unlike the super-little Higgs models where the soft masses are introduced by hand, all the soft masses in our model are calculable. Our model contains new gauge bosons and fermions at 1 TeV that are also present in other little Higgs models. Furthermore, our model predicts additional electroweak triplets and doublets at 1 TeV from the psedu-moduli in the meta-stable SUSY breaking sector.
The paper is organized as follows. In Sec. II we will briefly review the ISS model, its symmetries and its field content. In Sec. III we present our model and describe how to break to . In Sec. IV we explain how to break the electroweak symmetry. We address the vacuum alignment and order one quartic Higgs coupling issues. We also explicitly minimize the full effective scalar potential to support our step-by-step analysis. In Sec. V we discuss the soft masses of gauginos and sferminos generated via gauge interactions. In Sec. VI we provide a sample of the mass spectrum of our model. We conclude in Sec. VII.
Ii Meta-stable SUSY breaking
The ISS model is a deformed supersymmetric QCD, with massive fundamental flavors. is taken to be in the free magnetic range, for a controllable IR description of the theory. For concreteness, we will concentrate on the simplest model where the magnetic gauge group is trivial.
In terms of the superfields with normalized kinetic terms
the low-energy effective superpotential we study is
where is a dimensionless parameter of order one; is related to a holomorphic scale of the microscopic theory, but much smaller than . are identified as mesons, baryons and antibaryons of the electric theory. In addition, there is an instanton generated operator . For , this term is irrelevant and will be neglected in discussions of the physics around the origin of . Thus, below we will set . The reason for this choice will be discussed later.
The F-terms of the meson field are
SUSY is broken since has rank one while has rank . Up to global transformations, the vacua are
In these vacua, the global symmetry is broken to .
To see what the light fields are, we expand around Eq. (3) using the following parametrization
where the component fields transform under the unbroken global symmetry as
and has a non-zero F-term
Another two fields, corresponding to the Cartan generators in the adjoint representation, have nonzero F-terms as well.
To identify the Goldstone fields, we use the following non-linear parametrization of scalar fields
To the leading order, and are related to , as
The scalar mass spectrum is
where is the heavy combination of the triplets , while the massless combination together with the singlet are the seven Nambu-Goldstone bosons (NGB)’s in the coset space . are pseudo moduli and obtain masses of order through the one-loop Coleman-Weinberg potential
In the UV regime, the instanton term in the superpotential can not be neglected and is crucial to generate supersymetric vacua . The distance in field space between the local vacua and the global minima is controlled by a small parameter , and thus the meta-stable vacuum can have a lifetime much longer than the age of the Universe.
Iii The model
The ISS model provides a way to break SUSY and can be used to construct a gauge mediation model, where the soft masses of gauginos and sfermions are roughly of the same scale, . Here, are the gauge coupling constants in the standard model, is the value of an F-term indicating SUSY breaking and is the supersymmetric mass of the messenger. As the superpartner masses are of order 100 GeV or more, we need to have TeV. To avoid tachyonic directions of the messenger fields, and the lowest allowed scale of is of order .
Since the Higgs boson mass is likely to be , it has to be at least two-loop factor below the SUSY breaking scale in the ISS model. Below the SUSY breaking scale, we introduce the collective symmetry breaking mechanism as in little Higgs models to protect the mass of the Higgs boson. In the simplest little Higgs model Schmaltz:2004de (), the Higgs boson mass is lighter than the ’s breaking scale TeV by a factor of . For our purpose, we need to achieve the ’s breaking scale one loop factor lower than the SUSY breaking scale TeV.
To achieve this purpose and to have enough light degrees of freedom containing the Higgs doublet as a PNGB, we adopt two ISS sectors with global symmetry. For simplicity, we choose two identical ISS sectors by imposing a symmetry between them. In each ISS sector, the global symmetry and SUSY are broken at the same scale, , where the coupling in the ISS sectors is chosen to be of order one. Then there are two massless triplets and , which are the NGB’s in each sector. Hence, the effective field theory below resembles the simplest little Higgs model with the unbroken global symmetry . After gauging the diagonal subgroup, and become PNGB’s. We will later introduce a superpotential to spontaneously break the approximate global symmetry to . The Higgs doublet is a PNGB and contained in the light triplets as
where and are broken at the same scale ; is chosen to have properly normalized kinetic term of the Higgs doublet.
iii.1 Higgs sector and D-term
For the two identical ISS sectors, the superpotential is
The symmetry breaking and the vacua in both sectors are the same as described in section II.
The parametrization around the ISS vacua are then
where . The unbroken global symmetry is . The diagonal subgroup of this global symmetry is gauged and denoted as with the electroweak gauge group as a subgroup. Under the gauge symmetries, the field content in the Higgs sector is described in Table 1
The D-term potential of preserves the global symmetry, so it will not give a potential to the Higgs doublet and we neglect it here. To study the Higgs doublet potential, we ignore fields , and and only consider the following part of the D-term potential
where in the second line we expand the D-term potential in terms of , which are defined in section II; is the gauge coupling; are the gererators of and ; the dots represent other preserving terms and do not contain quartic terms of and . Substituting , the D-term does not provide a self-interaction potential of and the Higgs doublet at tree level (the Higgs doublet is embedded in as in Eq. (10)). Actually, this result comes from the same vacuum expectation values (VEV)’s of and in the ISS model. After integrating out the heavy modes, the one-loop effective potential of the two light triplets from the gauge interaction is of the form
where order one numbers are neglected in the parenthesis. Substituting the parameterization of and in Eq. (10) into this potential, the Higgs doublet mass is and it is at most of order 100 GeV.
iii.2 Breaking to
To spontaneously break the global symmetries, we employ the trick of adding vector-like fermions that has been widely used in the little Higgs model building. The superpotential we propose is
where the superfields and with . Here and are top and bottom left-handed quarks. The charge assignments of those new fields are listed in Table 2.
We only list fields in the top sector, which provides the dominate contribution to the Higgs doublet mass. The full list of fields without gauge anomalies can be found in Roy:2005hg ().
In order not to change the “rank condition” of the ISS model and not to generate VEV’s for the charged fields, the following relation among Yukawa couplings must be satisfied
Other than the superpotential in Eq. (15), we also need an additional superpotential to generate order one quartic couplings for and . We introduce extra singlets ’s which couple to and as
where is of order one. This superpotential is invariant and contains operators and at tree level. Additional operators like from this superpotential do not provide a self-interaction potential of at tree level, but they contribute to the masses of at one-loop level. We will neglect their contributions by choosing to be smaller than the Yukawa couplings in Eq. (15).
From Eq. (15), we calculate the one-loop Coleman-Weinberg potential for those light triplets and . Together with the order one quartic couplings, the potential of those two triplets is
The fermion and scalar mass matrices are too complicated to be diagonalized analytically, and hence we present a numerical study here. Choosing , (satisfying the condition in Eq. (16)) and , we plot the triplet potential along the direction in Fig. 1.
Fig. 1 shows that the triplet develops a nonzero VEV at , which parametrically is
and breaks to . A similar result can be obtained for to break to .
The scale on the vertical axis of Fig. 1 comes from the product of the one-loop suppression factor and small values of near the minimum.
The potential of is periodic as are PNGB’s. It is found that the minimum in Fig. 1 is the global one, so we only consider the parameter space near the origin.
If the operator were generated in our effective potential, naive dimensional analysis suggests it would give an contribution to the Higgs doublet mass. However, our superpotential preserves the symmetry which forbids the term.
Logarithmic divergence for the one loop potential of the triplets from Eq. (15) is absent. This is due to the double protection of SUSY and the collective symmetry breaking mechanism on the masses of PNGB’s or triplets here. Without SUSY, more than one Yukawa coupling is needed to generate a potential for PNGB’s, and then there is not any quardratic divergent potential for the PNGB’s at one loop. SUSY protects the mass of PNGB’s furthermore and leaves us one finite potential for those triplets at one loop.
After and develop nonzero VEV’s as (the alignment issue will be discussed later), the global symmetry spontaneously breaks to and generates two doublets as NGB’s. The gauge symmetry spontaneously breaks to . One of those two NGB doublets is eaten by the gauge bosons, ’s and , and the other one becomes a PNGB, which is the Higgs doublet in our model and parameterized in Eq. (10). The D-term also gives an additional mass to one linear combination of and to complete the “super Higgs mechanism”.
Iv Electroweak symmetry breaking
Other than providing negative mass terms to and , the one-loop Coleman-Weinberg potential of calculated from Eq. (15) also contains an operator, , with a positive coefficient. For example, using the same values of Yukawa couplings as in the previous section, the coefficient is . Substituting the parametrization of and in terms of the Higgs doublet in Eq. (10), we have
Here . The Higgs doublet mass is negative and the electroweak symmetry successively breaks to . However, this result is based on the assumption that the VEV’s of and are aligned (the VEV’s of both triplets have only nonzero values at the third entries). Therefore, we need to address the vacuum alignment issue first.
iv.1 Vacuum alignment
Although the operator with a positive coefficient gives a negative mass to the Higgs doublet, it prefers anti-aligned VEV’s of and . Thus, we need to introduce a new operator together with to achieve the alignment and a negative mass for the Higgs doublet at the same time. One possibility is to include an operator, , which is the only operator explicitly breaking one linear combination of . Therefore, the coefficient can be as small as possible and we choose it to be . Combining the -term potential of and , we have breaking parts of the potential as
where we define and the positive coefficient in front of as . For this potential, under the following condition
the VEV’s of and are aligned and at the same time the Higgs doublet has a negative mass
iv.2 Order one quartic Higgs coupling
To obtain the electroweak symmetry breaking scale at 246 GeV, we need to introduce an order one quartic Higgs coupling, which is difficult without generating a large Higgs mass term. This problem is present not only in a realistic super little Higgs model but also in the simplest little Higgs model. In our model, we use the “sliding singlet mechanism” slide () to reach this goal by including the following renormalizable superpotential:
By assigning appropriate charges to and , this superpotential also preserves the global symmetry. The -term potential of the ’s is
Minimizing this potential, one can see that the VEV’s of cancel the VEV’s of . After substituting the parametrization of in Eq. (10), we have a potential of the Higgs doublet without any VEV’s. Hence, the quartic coupling of the Higgs doublet is of order one, and with a negative mass from Eq. (23) the Higgs doublet develops a VEV at .
iv.3 Complete triplet potential
Up to now, we have first studied the spontaneously breaking of to , and then studied the Higgs doublet potential to break the electroweak symmetry. On the other hand, we can also study the full invariant potential of the two light triplets , minimize their potential and derive the final VEV structure. Combining Eqs. (18), (21), (25) and choosing (the reason to choose a smaller than the Yukawa couplings in Eq. (15) is the same as ), we have the following complete potential
with and determined by the Yukawa couplings in Eq. (15). The crucial “–” signs in the first two terms and the “+” sign before the fourth term are also derived from Eq. (15). Here, we neglect the contributions to the effective potential from the gauge interaction, due to the smallness of the gauge coupling compared to the Yukawa couplings. For simplicity, we neglect the quartic operator generated at one loop, which does not change our final result significantly.
Minimizing this potential, we derive
Appoximately we have
Eqs. (23), (29) show a relation between the Higgs doublet mass and its VEV as . Using GeV, the Higgs boson mass is around 150 GeV. Actually the ratio of the Higgs boson mass over depends on order one Yukawa couplings in our model. Thus, the Higgs boson mass can vary from 100 GeV to 200 GeV by choosing from to .
After electroweak symmetry breaking, the lightest quark in the top sector, which is identified as the top quark in the standard model, has top Yukawa coupling
Choosing the same numerical values as in Fig. 1, .
V Direct gauge mediation
In our model, the soft masses of gauginos and sfermions arise from gauge mediation. The messenger fields are embedded inside the SUSY breaking sector, and therefore our model belongs to a direct gauge mediation model.
The couplings in the superpotential in Eq. (15) explicitly break the continuous symmetry to parity. We calculate the one-loop Coleman-Weinberg potential of the pseudo-moduli , and find that they develop nonzero VEV’s as (similar results can be found in R symmetry breaking ()). In our model, it is crucial to have non-zero VEV’s of to generate masses for and gauginos, which include winos and bino. But the masses of gluinos can be generated independent of .
For the gluino masses, , , and , are the “messengers” in our model. We only need to consider the last four operators in Eq. (15)
where . The SUSY breaking mass terms of the messenger fields are given as . From the standard one-loop Feynman diagram calculation, we have the formula for the masses of gluinos as
where the factor of 2 comes from the presence of two ISS sectors and the function is defined as
Here, are fermion masses and are scalar masses. Corresponding expressions are
Three interesting limits for Eq. (32) can be studied:
When or , we have and . As , the gluinos mass is zero. For a small , expanding the gaugino mass in terms of , we find that the leading non-vanishing term is proportional to which agrees with Izawa:1997gs (). In terms of the fermion masses, the leading term of gluino masses is
When , this is the simpliest case of gauge mediation models with only one messenger. In this case, , , , and . After algebraic manipulations, the masses of the gluino are , with . This agrees with the result in the literature Martin:1996zb ().
When , the continuous symmetry is unbroken. We have , and . Considering that , we have .
In our model, all the Yukawa couplings are generally order one numbers, so and are at the same scale. With and , we have