aa r X i v : . [ h e p - ph ] O c t A Simple Model of Direct Gauge Mediation
Sibo Zheng and Yao Yu
Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Abstract
In the context of direct gauge mediation Wess-Zumino models are very attractive in su-persymmetry model building. Besides the spontaneous supersymmetry and R -symmetrybreaking, the problems of small gaugino mass as well as µ and B µ terms should be solvedso as to achieve a viable model. In this letter, we propose a simple model as an existenceproof, in which all these subjects are realized simultaneously, with no need of fine tuning.This completion also implies that much of parameter space for direct gauge medition canbe directly explored at LHC.08/ 2011upersymmetry (SUSY) is an appealing candidate for explaining the mass hierarchyand providing the unification of gauge couplings. Experimental searches at colliders suchas LHC give arise to strigent constraints on supersymmetric physical parameters at lowenergy where SUSY must be broken. Some difficulties can be avoided by adjusting themechanism of SUSY breaking or allowing a few fine tunings.On the realm of SUSY model building, gauge mediated SUSY breaking , is one of themost well studied scenarios for a few reasons. At first, the problem of flavor changingneutral currents can be naturally solved with supersymmetric particles ∼ W = f X + ( λ ij X + m ij ) ϕ i ˜ ϕ j + · · · (1)where neglected terms denote the cubic terms.What is of more interest is to directly apply these O’ Raifeartaigh models to modelsbuildings in the context of gauge mediation, i.e, direct gauge mediation (DGM) [4, 5, 6].In contrast with the minimal gauge mediation (see review [7] and references therein), the isno need to introduce additional messenger sector in DGM. At first sight, it is observed thatthe gaugino mass of order O ( F ) ( √ F refers to the supersymmetry breaking scale) oftenvanishes in direct gauge mediated O’ Raifeartaigh models, Now it is understood [3] thatthis phenomena is tied to the global vacuum structure composed of pseudomoduli space X , whether R -symmetry is spontaneous broken or not. In light of this new finding, variousO’ Raifeartaigh models where the gaugino mass problem can be resolved are proposed[8, 9, 10, 11, 12, 13, 14, 15, 16], some of which even have microscopic completions [10, 16].In this letter, we discuss another important subject left in DGM, that is the generationof viable µ and B µ terms as in ordinary gauge mediation [17, 18, 19, 20, 21, 22, 23] . In [22], the authors discuss the strongly coupled generation of µ term in context of direct gauge R -symmetry, and then address the gaugino mass and µ problem simultaneously.First, what kind of O’ Raifeartaigh models in (1) can solve the gaugino mass prob-lem ? We can take a few limits in (1) for illustration. If λ is diagonal (via bi-unitarytransformation ) and m = 0, this actually reduces to the minimal gauge mediation, inwhich det M = X N det λ . As well known there is no small gaugino mass problem in thiscontext . However, it is nerve considered as starting point of direct gauge mediation,as spontaneous SUSY breaking can not be realized in this setup. If m is diagonal thendet M =det m , which results in spontaneously broken SUSY and vanishing gaugino massat order of O ( F ).Therefore, in order to render the O’ Raifeartaigh model to generate the one-loopgaugino masses, or equivalently guarantee the determinant to depend on X , there mustbe at least one non-zero diagonal element in λ . So the superpotential can be constructedas the mixing of those of minimal setup and tree-level mass terms of messengers.Now, we consider a concrete model in light of above observations, whose superpotentialis given by, W = f X + λ X (cid:16) S ˜ S + T ˜ T (cid:17) + m S ˜ T + λ X ( ϕ ˜ ϕ + ϕ ˜ ϕ ) + m ϕ ˜ ϕ + λ X ( ϕ ˜ ϕ + ϕ ˜ ϕ ) + m ϕ ˜ ϕ (2)This is the minimal setup as we will find. We assume all the masses and couplings in(2) are real without loss of generality. The couplings in (2) can be realized via imposingglobal symmetries [ SU (2) × SU (2)] as follows,Φ = ϕ ϕ ! , ˜Φ = ˜ ϕ ˜ ϕ ! , Σ = ϕ ϕ ! , ˜Σ = ˜ ϕ ˜ ϕ ! . (3)Also the global symmetry assignment results in the degeneracies λ = λ and m = m .Thus, eq(2) can be rewritten as, W = f X + λ X (cid:16) S ˜ S + T ˜ T (cid:17) + m S ˜ T + λ X (cid:16) Φ ˜Φ + Σ ˜Σ (cid:17) + m Φ ˜Σ (4) mediation. The field space composed of the pseudomoduli X is stable globally in this type of O’ Raifeartaighmodels, which implies that the determinant M is a constant [3].
2e assume S, ˜ S and T, ˜ T as standard model singlets. Additional coupling associatedwith the Higgs fields can be introduced when the SU (2) global symmetries are directlygauged as the standard model electroweak groups, W = λ µ ˜ S ˜Φ H µ + λ d S Φ H d (5)In particular, either S, ˜ S fields or T, ˜ T can couple to the Higgs doublets, but they can notbe allowed to appear in (5) at same time as a result of R -symmetry. Similarly, eitherΦ , ˜Φ or Σ , ˜Σ can be coupled to Higgs fields. We choose the set in (5) for example. Thesuperpotential of O’ Raifeartaigh models we consider is W = W + W , which respectsgauge symmetries of standard model and R -symmetry involved. Once all the subjectsinvolved in SUSY model buildings are realized in such kind of models, one can add tripletfields of QCD gauge group in (2) so as to complete the model.According to (2) and (5), the vacuum is represented by, S = ˜ S = 0 , ΦΣ ! = 0 , ˜Φ˜Σ ! = 0 , X arbitrary (6)with potential V = f . To achieve this vacuum, the property that there are diagonal λ andnon-zero mass terms in (2) is crucial in above analysis. At this SUSY breaking vacuum (6)the gauge symmetries of standard model is unbroken. Even without studying the detailsof pseudomoduli space X , one expects that there is no gaugino mass problem in thismodel. Since in the region X →
0, some freedoms in messengers become tachyonic. Thismeans that the vacuum (5) is not stable globally. From (2), the eigenvalues of messengerfermion mass squared M F are given by, m / ,i ± = m
12 + x i ± r
14 + x i ! (7)for a given basis i . In (7) we have defined the dimensionless coefficients x i = λ i X/m i .Similarly, it is straightforward to evaluate the messenger boson mass squared M B . Fromthese eigenvalues we verify that some fermions are massless while some bosons tachyonicat small X < √ f . So the physical parameter space is given by p f << X < min ( m i ) (8) The reason is due to the absence of quardratic mass terms for S and T in (2). We undersatandthis fact as a consequence of R ( S ) = 1 and R ( T ) = 1. Similar understanding can also be applied tosingelt fields ˜ S and ˜ T . O (10) for the first constraint in (8) is sufficient to guarantee thepositive masses of messenger fields. In this note, we will take the small F limit in orderto simplify the analysis of Coleman-Weinberg potential in the next paragraph.Let us examine the R -symmetry breaking in our model. In (2) one finds that theremust be R -charge assignments other than 0 or 2 in (2). Following the argument in [24],which states that R -symmetry can not be broken except there are fields with R -chargeother than 0 and 2, one can see that the R -symmetry breaking or equivalently negativemass squared m X is not difficult to be realized. According to discussions in the previousparagraph, it is sufficient to study the region of moderate X value, we will focus onthis region with small F -term. Under limit (8) the one-loop Coleman-Weinberg potential V CW ( X ) for the pseudomoduli at moderate X is approximately given by, V CW ( X ) = 5 f π i =3 X i =1 λ i V ( x i ) , x i = λ i X/m i (9)where V ( x i ) = −
21 + 4 x i + 4 log x i + 2 x i + 1(4 x i + 1) log 2 x i + 1 + p x i + 12 x i + 1 − p x i + 1 (10)The Coleman-Weinberg potential is plotted in fig. 1; one finds that V CW is minimized at x = x = x ≃ .
25 or X ∼ . V C W Figure 1: V CW varies as function of x i in unit of f . For illustration, take the particularvalues λ = λ = λ = 3 and m = m = m = m.Firstly, we set all masses m i are unified in order to simplify the analysis. As shownin fig. 2( a ), if one wants to obtain X = 0 . m , λ and λ should be chosen around the4 .5 1. 1.5 2. 2.5 3.0.51.01.52.02.53.0 H a L Λ Λ H b L Λ =Λ m (cid:144) m = m (cid:144) m Figure 2: Unified masses m i =m and X = 0 . a ). The parameter space composedof λ and λ is shown in the region 0 . ≤ λ ≤
3. Non-degenerate masses among m and m in ( b ). We set the messenger scale X = 0 . m , while λ varies in the region 0 . − . − .
5, when λ varies from 0.1 to 3 . If X ≤ .
01 m, we find that there areno parameter space allowed. Relax the condition m i = m and allow deviation of m from m , we show in fig. 2.( b ) the parameter space when λ = λ , X = 0 . m and λ variesfrom 0.1 to 3.Now we proceed to discuss the soft terms induced by superpotential (5), which can beread from the one-loop effective Kahler potential K eff after integrating out the messengerfields involved [27], K eff = − π T r (cid:18) M † M log M † M Λ (cid:19) (11)From (5), in the case m = m and λ = λ matrix M † M is reduced to 4 × M † M = λ | X | + λ µ | H µ | λ λ µ X ∗ H µ + λ λ µ XH ∗ µ λ m X ∗ λ λ d XH ∗ d + λ λ µ X ∗ H µ λ | X | + λ d | H d | + m λ m X λ d m H ∗ d λ m X ∗ λ | X | λ m X λ d m H d λ | X | (12)under basis (cid:16) Φ , ˜ S, ˜ T , Σ (cid:17) M (cid:16) ˜Φ , S, T, ˜Σ (cid:17) T . Large Yukawa couplings often give rise to the problem of Landau pole in the context of direct gaugemediation [25, 26] µ = ∂∂ ¯ θ Z µd | θ =¯ θ =0 ,Bµ = − ∂∂ ¯ θ ∂∂θ Z µd | θ =¯ θ =0 m H µ = − ∂∂ ¯ θ ∂∂θ log Z µ | θ =¯ θ =0 (13) m H d = − ∂∂ ¯ θ ∂∂θ log Z d | θ =¯ θ =0 where Z µd and Z µ,d are given by, Z µd = ∂∂ ( H µ H d ) K eff | H µ = H d =0 ,Z µ,d = ∂∂ ( H † µ,d H µ,d ) K eff | H µ,d = H † µ,d =0 (14)Since these soft terms are generated through one hidden sector in our framework, ourmodel belongs to what is known as one-scale gauge medaition. As discussed in [28], oneroughly expects a relation as | Bµ |∼ m H µ,d >> µ (15)which plagues these one-scale models and indicates the failure of EWSB. However, moreprecise estimates needs to be done so as to verify this relation given a specific model, andit is not impossible to avoid this relation in some circumstances.Here we point out some possibilities. One choice is that m H µ is negative, with itsabsolute value smaller than positive m H d but larger than µ . Another choice is that oneallows a large m H d and small m H µ , with a small hierarchy m H d >> Bµ so that it canbalance the influence coming from the small hierachy Bµ >> µ [31]. We refer [30] to thereads for more discussions about this issue. As we will see the model we discuss here is anew example in the first choice.Since the matrix (12) is quite complicated so that the effective Kahler potential cannot be generally evaluated, we take the limit m = m and λ = λ to simplify thesimulation. Note that these choices correspond to a favored parameter space, as seen infig 2.( b ).The leading contributions to m H µ,d are composed of two parts. One arises from theordinary gauge mediation. The other comes from the superpotential (5). The later6ontribution induced at one-loop, generally dominates over the former. By using theconditions of electroweak symmetry breaking,( c.
1) : ( Bµ ) > ( | µ | + m H µ )( | µ | + m H d )( c.
2) : 2
Bµ < | µ | + m H µ + m H d (16)For λ = λ = 1 and fixed scale X = 0 . m , it turns out that the allowed parameterspace is given by , λ ∼ . , m /m ∼ .
12 (17)which results in the following spectra in our model, m H µ : µ : Bµ : m H d ∼ (18)after we put values of (17) into (12) and (13). The spectra (18) suggests that our modelis an example of large m H d and small m H µ mentioned above.What about the RG effects on the spectra given by (12) when one runs from X tothe electroweak scale ? Since there are no multiple messenger threshold corrections in ourmodel, the RG effects are quite simple. According to the RG equations of MSSM given in[29], one observes that the m H µ receive its quantum corrections more substantially than µ , m H d and B µ . If we restrict us to low-scale gauge mediation with X ∼ − TeV,the correction can be estimated through linear approximation. For the spectra given by(18) , δm H µ ∼ − . × m H d / π ∼ − µ . This negative contribution implies that the firstcondition in (16) can be still satisifed, while the second condition does not substantiallymodified . We refer the readers to the recent work [30] on this subject through effectivefield theory analysis.What about the other choices such as m = m << m and λ = λ << λ , or m = m << m and λ = λ >> λ , or m = m >> m and λ = λ << λ ? We findthat it is often impossible to both satisfy the electroweak symmetry breaking conditions( c.
1) and ( c.
2) in these cases. What is worse is that the parameter space to generate theone-loop gaugino masses is substantially suppressed under these limits, as shown in fig.2( b ). Since couplings λ µ and λ d are overall coefficients in µ and Bµ terms, we have taken λ µ = λ d = 1 forsimplicity. Also note that large deviation from λ ∼ λ ∼ X ∼ .
7n summary, we propose a simple Wess-Zumino model, which can serve as viable SUSYmodel of direct gauge mediation. In this scenario, all messengers involve in supersymmetrybreaking. The R -symmetry is also spontaneously broken as a result of the specific choicesof R -charges. Phenomenologically, We find the gaugino mass is induced at one-loop, withthe same order of the scalar masses. Also, there is no µ problem associated with softmasses in Higgs sector, which can be naturally solved in our model, with no need offine tunings among Yukawa couplings in the SUSY breaking hidden sector. Since most ofsupersymmetric particles ∼ Acknowledgement
We thank Jia-Hui Huang for discussions and Jin Min Yang for reading the manuscript.This work is supported in part by the Fundamental Research Funds for the CentralUniversities with project number CDJRC10300002.
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