# Effective Natural Supersymmetry from the Yukawa Deflected Mediations

Authors: Tai-ran Liang; Bin Zhu; Ran Ding; Tianjun Li
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

The natural supersymmetry (SUSY) requires light (≤1 TeV) stop quarks, light sbottom quark, and gluinos. The first generation of squarks can be effectively larger than several TeV which does not introduce any hierarchy problem in order to escape the constraints from LHC. In this paper we consider a Yukawa deflected mediation to realize the effective natural supersymmetry where the interactions between squarks and messengers are made natural under certain Froggatt-NielsenU(1)X charges. The first generation squarks obtain large and positive contribution from the Yukawa deflected mediation. The corresponding phenomenology and sparticle spectra are discussed in detail.


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## Body

## 1. Introduction

Gauge Mediated SUSY Breaking (GMSB) [1] is an elegant framework. In its minimal form, the SUSY breaking hidden sector can be communicated with visible sector only through usual gauge interaction. It can be realized by introducing spurion field X with X=M+θ2F and messenger fields Φ and the corresponding superpotential is written as(1)W=XΦΦ¯.Here spurion X couples to the SUSY breaking sector and X parameterizes the SUSY breaking effects and Φ are charged under the Standard Model (SM) SU(3)×SU(2)×U(1) gauge group. Since the mass matrix of scalar messenger components is not supersymmetric, the SUSY breaking effects from hidden sector can be mediated to visible sectors via messenger loops. Compared with gravity mediated SUSY breaking, GMSB has two obvious advantages: (i)
Soft terms are fully calculable. Even in the case of strongly coupled hidden sector, the soft terms can be still expressed as simple correlation functions of hidden sector, namely, the scenario of General Gauge Mediation (GGM) [2].(ii)
It is inherently flavor-conserving since gauge interaction is flavor-blinded and thus is strongly motivated by the SUSY flavor problem.However, the status of minimal GMSB has been challenged after the discovery of SM-like Higgs boson with a mass of 125 GeV [3, 4]. In order to lift Higgs mass to such desirable range, it then implies that Higgs mass should receive significant enhancement either from radiative corrections via stop/top loops [5, 6] or from extra tree-level sources [7]. The first option can be achieved through extremely heavy and unmixed stops or through lighter stops with maximal mixing (large trilinear soft term of stops) [8–11], while in minimal GMSB, the vanishing trilinear soft term at the messenger scale makes maximal mixing be impossible. The second option requires the extension of Minimal Supersymmetric Standard Model (MSSM) and has been widely investigated [12–27]. In this paper, we consider the first option where large trilinear term is required to soften fine-tuning. In fact, if the messenger sector is allowed to couple with squark or Higgs, the problem is improved with trilinear soft terms generated by additional interactions. The interactions can be generally divided into two types, that is, Higgs mediation and squark mediation. However, Higgs mediation generates irreducible positive contribution δmHu2~AHu2 and leads to large fine-tuning, which is the so-called A/mHu2 problem. The situation is quite different in squark mediation since it does not suffer from such problem and it allows better control to fine-tuning. One compromise is that squark mediation reintroduces dangerous flavor problem since there is no prior reason to specify the hierarchy and alignment of Yukawa matrix of squark. In this direction, Froggatt-Nielsen (FN) mechanism [28] is adopted as a canonical solution. Here we take the same strategy for squark mediation. In a previous study, [29] considered the type of sfermion-sfermion-messenger interaction with FN mechanism. In this work, we extend the model to include sfermion-messenger-messenger interaction and to examine its phenomenology systematically.The rest of this paper is layout as follows. In Section2, we present our notation and model contents. The realization of FN mechanism in Supersymmetric Standard Models (SSMs) and SU(5) models are reviewed in Section 3. In Section 4, the FN mechanism is extended to constrain the possible interactions between squarks and messengers. We show that a unique interaction can be obtained with appropriate charge assignment. In Section 5, we explore the phenomenology of this model with an emphasis on spectra and fine-tuning issues. The last section is devoted to conclusion.


## 2. Vector-Like Particles (Messengers) in the SSMs andSU(5) Models

First, we list our convention for SSMs. We denote the left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, right-handed neutrinos, and right-handed charged leptons asQi, Uic, Dic, Li, Nic, and Eic, respectively. Also, we denote one pair of Higgs doublets as Hu and Hd, which give masses to the up-type quarks/neutrinos and the down-type quark/charged leptons, respectively.In this paper, we consider the messenger particles as the vector-like particles whose quantum numbers are the same as those of the SM fermions and their Hermitian conjugates. As we know, the generic vector-like particles do not need to form completeSU(5) or SO(10) representations in Grand Unified Theories. In particular it does not need complete multiplets (GUTs) from the orbifold constructions [30–41], intersecting D-brane model building on Type II orientifold [42–44], M-theory on S1/Z2 with Calabi-Yau compactifications [45–47], and F-theory with U(1) fluxes [48–57] (For details, see [58]). Therefore, we will consider two kinds of supersymmetric models: (1) the SSMs with vector-like particles whose U(1)X charges can be completely different; (2) the SU(5) models.In the SSMs, we introduce the following vector-like particles whose quantum numbers underSU(3)C×SU(2)L×U(1)Y are given explicitly as follows:(2)XQ+XQc=3,2,16+3¯,2,-16;XU+XUc=3,1,23+3¯,1,-23;XD+XDc=3,1,-13+3¯,1,13;XL+XLc=1,2,-12+1,2,12;XE+XEc=1,1,-1+1,1,1.In theSU(5) models, we have three families of the SM fermions whose quantum numbers under SU(5) are (3)Fi=10,f¯i=5¯,where i=1,2,3 for three families. The SM particle assignments in Fi and f¯i are (4)Fi=Qi,Uic,Eic,f¯i=Dic,Li.To break the SU(5) gauge symmetry and electroweak gauge symmetry, we introduce the adjoint Higgs field and one pair of Higgs fields whose quantum numbers under SU(5) are(5)Φ=24,H=5,H¯=5¯,where H and H¯ contain the Higgs doublets Hu and Hd, respectively.We consider the vector-like particles that form completeSU(5) multiplets. The quantum numbers for these additional vector-like particles under the SU(5)×U(1)X gauge symmetry are(6)XF=10,XF¯=10¯,Xf=5,Xf¯=5¯.The particle contents from the decompositions of XF, XF¯, Xf, and Xf¯ under the SM gauge symmetries are(7)XF=XQ,XUc,XEc,XF¯=XQc,XU,XE,Xf=XD,XHu,Xf¯=XDc,XHd.Here we have introduced two pairs of Xf and Xf¯, and we denote them as Xfi and Xf¯i with i=1,2.In this paper, we consider the messenger parity. We further consider the messenger parity, for example, discreteZn symmetry with n≥2. Under this Zn symmetry, the vector-like particles XΦ and XΦc transform are as follows:(8)XΦ⟶ωXΦ,XΦc⟶ωn-1XΦc,where ωn=1. Thus, the lightest messenger will be stable. If the reheating temperature is lower than the mass of the lightest messenger, there is no cosmological problem. This is indeed work in our models. Otherwise, we can break the messenger parity a little bit by turning on tiny VEVs for XL and/or XLc.In the gauge mediation, it is very difficult to obtain the Higgs boson with mass around 125.5 GeV due to the small top quark trilinear softAt term unless the stop quarks are very heavy around 10 TeV. To generate the large top quark trilinear soft At term, we introduce the superpotential term XQXUcHu [59, 60]. In addition, we consider high scale gauge mediation by choosing(9)S~1014GeV,FS~1020GeV.The point is that we can increase the magnitude of top quark trilinear soft term via RGE running. Another point is that the couplings between the spurion and messengers can be very small because FS/S2~10-8.


## 3. Froggatt-Nielsen Mechanism via an AnomalousU(1)X Gauge Symmetry

It is well known that the SM fermion masses and mixings can be explained elegantly via the FN mechanism, where an additional flavor dependent globalU(1)X symmetry is introduced. To stabilize this mechanism against quantum gravity corrections, we consider an anomalous gauged U(1)X symmetry. In a weakly coupled heterotic string theory, there exists an anomalous U(1)X gauge symmetry where the corresponding anomalies are cancelled by the Green-Schwarz mechanism [61–63]. For simplicity, we will not consider the U(1)X anomaly cancellation here, which can be done in general by introducing extra vector-like particles as in [64–67].To break theU(1)X gauge symmetry, we introduce a flavon field A with U(1)X charge -1. To preserve SUSY close to the string scale, A can acquire a VEV so that the U(1)X D-flatness can be realized. It was shown [64, 65] that(10)0.171≤ϵ≡AMPl≤0.221,where MPl is the reduced Planck scale. Interestingly, ϵ is about the size of the Cabibbo angle. Also, the U(1)X charges of the SM fermions and the Higgs fields ϕ are denoted as QϕX.The SM fermion Yukawa coupling terms arising from the holomorphic superpotential at the string scale in the SSMs are given by(11)-L=yijUAMPlXYUijQiUjcHu+yijDAMPlXYDijQiDjcHd+yijEAMPlXYEijLiEjcHd+yijNAMPlXYNijLiNjcHu,where yijU, yijD, yijE, and yijN are order one Yukawa couplings, and XYUij, XYDij, XYEij, and XYNij are nonnegative integers:(12)XYUij=QQiX+QUjcX+QHuX,XYDij=QQiX+QDjcX+QHdX,XYEij=QLiX+QEjcX+QHdX,XYNij=QLiX+QNjcX+QHuX.Similarly, the SM fermion Yukawa coupling terms in the SU(5) models are(13)-L=yijUAMPlXYUijFiFjH+yijDEAMPlXYDEijFif¯jH¯+yijNAMPlXYNijf¯iNjcH,where(14)XYUij=QFiX+QFjX+QHX,XYDEij=QFiX+Qf¯jX+QH¯X,XYNij=Qf¯iX+QNjcX+QHX.In addition, we shall employ the quark textures for the SSMs andSU(5) models in Table 1, which can reproduce the SM quark Yukawa couplings and the CKM quark mixing matrix for ϵ≈0.2 [64–66]. And the following lepton textures can reproduce the neutrino masses and PMNS neutrino mixing matrix:(15)YE~ϵcϵ4ϵ3ϵ1ϵ3ϵ2ϵ0ϵ3ϵ2ϵ0,MLL~Hu2Msϵ-5ϵ2ϵ1ϵ1ϵ1ϵ0ϵ0ϵ1ϵ0ϵ0,where c is either 0, 1, 2, or 3, and tan⁡β≡Hu/Hd satisfies ϵc~ϵ3tan⁡β. This neutrino texture requires some amount of fine-tuning as it generically predicts(16)sin⁡θ12~ϵ,Δm122~Δm232.Interestingly, with ϵ as large as 0.2, the amount of fine-tuning needed is not that huge and this is shown in the computer simulations of [64–66] with random values for the coefficients.Table 1
The quark textures in the SSMs andSU(5) models.
Yukawa
The SSMs
S
U
(
5
) models
Y
U
ϵ
8
ϵ
5
ϵ
3
ϵ
7
ϵ
4
ϵ
2
ϵ
5
ϵ
2
ϵ
0
ϵ
6
ϵ
5
ϵ
3
ϵ
5
ϵ
4
ϵ
2
ϵ
3
ϵ
2
ϵ
0
Y
D
ϵ
c
ϵ
4
ϵ
3
ϵ
3
ϵ
3
ϵ
2
ϵ
2
ϵ
1
ϵ
0
ϵ
0
ϵ
c
ϵ
4
ϵ
3
ϵ
3
ϵ
3
ϵ
2
ϵ
2
ϵ
1
ϵ
0
ϵ
0To be concrete, we choose theU(1)X charges for the SM fermions and Higgs fields in the SSMs as follows:(17)QQiX=3,2,0,QUicX=5,2,0,QDicX=c+1,c,c,QLiX=c+1,c,c,QEicX=3,2,0,QHuX=QHdX=0,with QϕiX≡(Qϕ1X,Qϕ2X,Qϕ3X) for the SM fermions ϕi.Also, we take the followingU(1)X charges for the SM fermions and Higgs fields in the SU(5) models:(18)QFiX=3,2,0,Qf¯iX=c+1,c,c,QHX=QH¯X=0.


## 4. Squark Mediation versus Higgs Mediation

Natural SUSY can be regarded as an effective SUSY scenario where only stop, gluino, and smallμ term are required in the spectra. As a consequence, the fine-tuning remains a manageable level. One nice property of Natural SUSY is that the first two generations of squarks can be very heavy without introducing any fine-tuning, which also evade bounds of SUSY direct searches from LHC. In terms of squark mediation with squark-messenger-messenger interaction, squarks receive additional positive contribution; thus it is possible to construct Natural SUSY model.The basic formulas to compute corresponding soft terms are given as(19)Aab=-132π2daijΔλaij∗λbijΛ,δmab2=1256π412dacBdBdeλacB∗λbcCλdeBλdeC∗+dacBdcdCλacB∗λbeBλcdCλdeC∗+dacBdbdCλacB∗λceBλdeC∗λbdC-dacddcfByacd∗ybdeλcfBλefB∗+12dacBdcefycefydef∗λacB∗λbdB+12dacddcefyacd∗ycefλbdBλefB∗+12dacBdBefλacB∗λefBybcdydef∗-2dacBCracBgr2λacB∗λbcBΛ2,where Λ=F/M and Crijk=cri+crj+crk is the sum of the quadratic Casimir of each field interacting through λijk. In above expressions, we do not include the contributions from usual GMSB (thus is labeled by δmab2) and all of indices are summed over except for a and b. Without the FN mechanism, there will be general interaction between Qi, Ui, and Di. The squark mediation is not automatically minimal flavor violation like Higgs mediation. The MSSM-MSSM mixing term gives rise to dangerous nonvanishing and nondiagonal soft masses; for example,(20)mQ1Q22~λq12λq22Λ2.The nondiagonal terms in (20) motive [68] to construct chiral flavor violation scenario where only single Qi, Ui, or Di is allowed to couple the messenger. As a result, the dangerous flavor violation term is suppressed naturally. However our situation does not belong to chiral flavor violation. In order to realize effective SUSY scenario, all the first and second generation squarks must be coupled to messengers in order to obtain large soft masses enhancement. It seems the nondiagonal term is inevitable in (20). The loop hole comes from the fact that the bound is greatly improved when the squarks are nondegenerate. In particular, the largest bound comes from the first generation squarks because of large PDF effect of first generation quarks. Therefore it strongly suggests us for considering the first generation squark mediation which is technically natural under FN mechanism. The FN natural model is free from MSSM-MSSM mixing and the formulas are reduced to(21)Aa=-116π2dacBλacB2Λ,δma2=1256π412dacBdBdeλacB2λdeB2+dacBdcdCλacB2λcdC2+dacBdadCλacB2λadC2-dacddcfByacd2λcfB2+12dacBdcefycef2λacB2+12dacddcefyacd∗ycefλadBλefB∗+12dacBdBefλacB∗λefByacdydef∗-2dacBCracBgr2λacB2Λ2.It is easy to demonstrate how FN mechanism makes the squark mediation flavor-blinded. The general squark-messenger-messenger interaction within the messenger sector beingSU(5) complete multiplets is divided into Q-type, U-type, and D-type mediations; here U and D denote u¯ and d¯, respectively. In Table 2, we list the complete messenger fields and their U(1) charge assignment.Table 2
Complete list of messenger fields and theirU(1) charge assignment.
Messenger
(
X
Q
,
X
Q
c
)
(XU,XUc)
(XL,XLc)
(
X
D
,
X
D
c
)
(XE,XEc)
XS
U
(
1
) Charge
(
3
,
-
3
)
(
-
5,5
)
(
2
,
-
2
)
(
3
,
-
3
)
(
0,0
)
0For theQ-type Mediation, the most general superpotential is(22)WQ=λq1iQiXQcXS+λq2iQiXDcXL+λq3iQiXUcXLc+λq4iXQXD,where i=1,…,3 is family indices. Based on Table 2, the Yukawa couplings in Q-type mediation can be determined as follows:(23)λq1i~1,1ϵ,1ϵ3,λq2i~ϵ2,ϵ,1ϵ,λq3i~ϵ6,ϵ5,ϵ3,λq4i~ϵ9,ϵ8,ϵ6.Terms with negative power of ϵ must be removed in order not to violate the holomorphic requirement of superpotential. While terms with positive order of ϵ can be ignored which is guaranteed by the smallness of ϵ. Therefore only λq11 is allowed under the consideration of FN mechanism and holomorphy. For now we only consider squark-messenger-messenger interaction; this is mainly because the squark-squark-messenger interaction under FN charges has been discussed in the literature [69]. Since only the Q1 mediation is allowed, there is no flavor-changing problem.For theU-type mediation the most general superpotential is(24)WU=λu1iUiXUXS+λu2iUiXDcXDc+λu3iUXQXLc+λu4iUiXEXD.According to FN mechanism the coupling looks like(25)λu1i~1,1ϵ3,1ϵ5,λu2i~1ϵ,1ϵ4,1ϵ6,λu3i~ϵ6,ϵ3,ϵ,λu4i~ϵ2,1ϵ,1ϵ3.It is similar to Q-type mediation; only λu11 is allowed. For D-type mediation we have(26)WD=λd1iDiXQXLc+λd2iDiXQcXQc+λd3iDiXDcXUc+λd4iDiXEcXU.Subject to the FN mechanism we obtain the couplings(27)λd1i~ϵ6,ϵ5,ϵ5,λd2i~1ϵ,1ϵ2,1ϵ2,λd3i~ϵ7,ϵ6,ϵ6,λd4i~1,1ϵ,1ϵ.Consequently the allowed Yukawa deflected mediation interaction for squarks is summarized as follows:(28)W=λqQ1XQcXS+λuUiXUXS+λdDiXEcXU.From (28), we obtain the extra contribution to soft masses for the first generation squarks. In other words there is no desirable large trilinear term At from (28) which motivates us to resort to Higgs mediation.Based on FN mechanism the only allowed superpotential for Higgs mediation is(29)WH=λhHuXDcXQ.It automatically preserves minimal flavor violation (MFV). Using (21), we obtain following soft terms:(30)At=-3Λλh216π2δmHu2=Λ218λh4-67g12/30+3g22/2+8g32/3λh2256π4,δmQ32=-3Λ2λh2yt2256π4,δmU32=-3Λ2λh2yt2128π4,δmQ12=Λ28λq4-2g12/30+3g22/2+8g32/3λq2256π4,δmU12=Λ25λu4-213g12/30+3g22/2+8g32/3λu2256π4,δmD12=Λ25λd4-214g12/15+8g32/3λd2256π4.The choice of Higgs mediation in (29) is crucial in reducing the fine-tuning:(i)
The trilinear soft term has an overall factor 3 coming from the higher representation ofSU(5). Thus it can give rise to large trilinear term compared with other Higgs mediation.(ii)
mHu2 has a negative contribution from SU(3) gauge coupling. Such a large coupling can reduce the fine-tuning easily.The parameter space is thus determined by the following parameters:(31)Λ,M,λq,λu,λd,λh,tan⁡β,signμ.


## 5. Phenomenology Analysis

In this section, we give a detailed discussion on the results from our effective supersymmetry model. In particular the Higgs mass, stop mass, gluino mass, and fine-tuning are given explicitly. In our numerical analysis, the relevant soft terms are firstly generated at messenger scale in terms of gauge mediation and Higgs and squark mediation. The low scale soft terms are obtained by solving the two-loop RG equations. For this purpose, we implemented the corresponding boundary conditions in (30) into the SARAH [70–74] package. Then SARAH is used to create a SPheno [75, 76] version for the MSSM to calculate particle spectrum. The tasks of parameter scans are implemented using package SSP [77].The framework that we concentrate on is MSSM with Yukawa deflected mediation. Its input parameters are given in (31). The scan range we adapt is (32)Λ∈6×104,6×105GeV,λh∈0,1.2.Other parameters are fixed to M=108 GeV, tan⁡β=10, and sign(μ)=1. For the parameters in squark mediation, we divide it into two scenarios:(33)Degenerated  squark:λq=λu=λd=0,Nondegenerated  squark:λq=λu=λd=1.2.In the scan, various mass spectrum and low energy constraints have been considered and listed as follows:(1)
The Higgs mass constraints:(34)123GeV≤mh≤127GeV.(2)
LEP bounds andB physics constraints:
1.6×10-9≤BR(Bs→μ+μ-)≤4.2×10-9(2σ) [78],2.99×10-4≤BR(b→sγ)≤3.87×10-4(2σ) [79],7.0×10-5≤BRBu→τντ≤1.5×10-42σ [79].(3)
Sparticle bounds from LHC Run-II:
(i)
Light stop massmt~1>850GeV [80, 81],(ii)
Light sbottom massmb~1 > 840–1000 GeV [82, 83],(ii)
Degenerated first two generation squarks (both left-handed and right-handed)mq~ > 1000–1400 GeV [83],(iv)
Gluino massmg~>1800GeV [81, 84].Finally, Barbieri-Giudice measure [85, 86] is used to quantify the fine-tuning:(35)ΔFT≡max⁡Δa,whereΔa≡∂log⁡mZ2∂log⁡a,where a denotes the input parameters in (31).In Figures1–3, we display the contour plots of important mass spectra and fine-tuning measure ΔFT in the [λh,Λ] plane. There are some notable features which can be learned from these figures and summarized as follows:(1)
The Higgs Mass. The Higgs mass range is taken from 123 GeV to 127 GeV in our numerical analysis. For small λh, one expects Higgs mass simply growth with an increases of Λ. When λh increases, the allowed parameter space is forced to shift to smaller Λ region in order to obtain correct Higgs mass.(2)
The Fine-Tuning Measure. For small values of Λ and λh, ΔFT is usually dominated by Λ. Since in these regions the RGE effects are most important, the contribution to the fine-tuning of λh, which only affects the boundary conditions, is negligible. The important parameter thus is Λ which sets the range of the RGE running. For moderate Λa and λh, the contributions from μ and Λ are almost comparable. When λh becomes large it is always the biggest contributor to fine-tuning measure independent of the value of Λ.(3)
The Squark and Gluino Masses. Both stop and gluino masses fall into multi-TeV range and therefore out of current LHC reach.Figure 1
Distribution of Higgs mass in[λh,Λ] plane.Figure 2
Distributions of stop (a) and gluino mass (b) in[λh,Λ] plane.
(a)
(b)Figure 3
Distribution of fine-tuning measure in[λh,Λ] plane.


## 6. Conclusions

In this paper, we investigated the extended gauge mediation models where Yukawa interaction between messengers and matter superfields is made natural under the consideration of F-NU(1) symmetry. Because of Higgs mediation the large A-term is generated naturally which can be used to enhance the Higgs mass efficiently. Considering the additional quark mediation, it is found that first generation squarks get large positive contribution, thus escaping from dangerous LHC constraints. We further study the parameter space and phenomenology numerically. The results show that the model is still promising under the stringent LHC constraint.



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