Eur. Phys. J. C (2011) 71:1811DOI 10.1140/epjc/s10052-011-1811-2

Regular Article - Theoretical Physics

Higgs as a pseudo-Goldstone boson, the mu problemand gauge-mediated supersymmetry breaking

Anna Kaminska1, Stéphane Lavignac2,a

1Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Hoza 69, Warsaw, Poland2Institut de Physique Théorique, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, Franceb

Received: 6 September 2011© Springer-Verlag / Società Italiana di Fisica 2011

Abstract We study the interplay between the spontaneousbreaking of a global symmetry of the Higgs sector andgauge-mediated supersymmetry breaking, in the frameworkof a supersymmetric model with global SU(3) symmetry. Inaddition to solving the supersymmetric flavor problem andalleviating the little hierarchy problem, this scenario auto-matically triggers the breaking of the global symmetry andprovides an elegant solution to the μ/Bμ problem of gaugemediation. We study in detail the processes of global sym-metry and electroweak symmetry breaking, including thecontributions of the top/stop and gauge-Higgs sectors to theone-loop effective potential of the pseudo-Goldstone Higgsboson. While the joint effect of supersymmetry and of theglobal symmetry allows in principle the electroweak sym-metry to be broken with little fine-tuning, the simplest ver-sion of the model fails to bring the Higgs mass above theLEP bound due to a suppressed tree-level quartic coupling.To cure this problem, we consider the possibility of addi-tional SU(3)-breaking contributions to the Higgs potential,which results in a moderate fine-tuning. The model pre-dicts a rather low messenger scale, a small tanβ value, alight Higgs boson with Standard Model-like properties, andheavy higgsinos.

1 Introduction

Among the proposed extensions of the Standard Model, su-persymmetry is one of the most attractive from a theoreticalpoint of view, in particular because it automatically solvesthe hierarchy problem. However, the lack of experimental

a e-mail: stephane.lavignac@cea.frbLaboratoire de la Direction des Sciences de la Matière du Commis-sariat à l’Energie Atomique et Unité de Recherche Associée au CNRS(URA 2306).

evidence put strong constraints on supersymmetric mod-els such as the Minimal Supersymmetric Standard Model(MSSM). The fact that no superpartner has been discov-ered so far implies a (at least in part) heavy supersymmetricspectrum, which exacerbates the “little hierarchy” problemassociated with the LEP bound on the Higgs mass and in-creases the level of fine-tuning in the Higgs potential. Fur-thermore, the absence of any significant deviation from theStandard Model predictions in flavor physics places strongrestrictions on the generational structure of squark and slep-ton masses. Gauge mediation [1–7] offers a natural solutionto this problem: supersymmetry breaking is communicatedto the observable sector by gauge interactions, and is there-fore automatically flavor blind. On the other hand, gaugemediation suffers from the so-called μ/Bμ problem [8],i.e. the fact that the μ and Bμ parameters of the MSSMare typically generated at the same loop order, leading toBμ � |μ|2, which is inconsistent with natural electroweaksymmetry breaking.

Both problems—the little hierarchy problem and theμ/Bμ problem of gauge mediation—may actually have acommon solution in terms of pseudo-Goldstone bosons.In Ref. [8], a mechanism involving additional singlet su-perfields was proposed to generate the μ and Bμ param-eters at the one- and two-loop levels, respectively, lead-ing to the order-of-magnitude relation Bμ ∼ |μ|2. It waspointed out that this relation has a pseudo-Goldstone in-terpretation: in some limit where the Higgs superpoten-tial becomes invariant under a global SU(3) symmetry,one combination of the two Higgs doublets remains mass-less after spontaneous breaking of this symmetry due tothe relation Bμ = |μ|2 (in which the soft Higgs mass pa-rameters have been omitted). Regarding the little hierar-chy problem, it is well known that it can be alleviatedif the Higgs boson arises as the pseudo-Goldstone bosonof some spontaneously broken approximate global symme-try, a scenario known as little Higgs [9–12]. It was shown

Page 2 of 14 Eur. Phys. J. C (2011) 71:1811

in Refs. [13–17] that the combination of supersymmetryand of a global symmetry leads to an improved protec-tion of the Higgs potential. Namely, the logarithmic depen-dence of the MSSM Higgs mass parameter on the cut-offscale is replaced by a dependence on the scale of sponta-neous global symmetry breaking, thus reducing the needfor a fine-tuning. Explicit realizations of this idea of dou-ble protection [15] of the Higgs potential by supersym-metry and by a global symmetry have shown that a fine-tuning smaller than 10% can be achieved with squark massesaround 1 TeV.

In this paper, we revisit these issues by combininggauge-mediated supersymmetry breaking with a sponta-neously broken global symmetry in the Higgs sector. Weconsider a simple model with a global SU(3) symme-try, which is essentially a gauge-mediated version of themodel of Ref. [15], and study it in detail. We pay partic-ular attention to the spontaneous breaking of the globalsymmetry, and check whether the SU(3)-breaking mini-mum is the global minimum of the scalar potential. A keyrole in the process of global symmetry breaking is playedby the tadpole of an SU(3) singlet field, which is gener-ated by loops of messenger fields together with the sin-glet soft terms. This tadpole also triggers the generationof the μ and Bμ parameters; however, they do not con-tribute to the potential of the lightest Higgs doublet due toits (pseudo-)Goldstone boson nature. As a result, the hig-gsinos and the non-Goldstone Higgs scalars can be madeheavy with masses of order |μ| � MZ without introducinga strong fine-tuning in the Higgs potential. We then computethe one-loop effective potential of the pseudo-GoldstoneHiggs doublet and study electroweak symmetry breaking.While the corrections to the pseudo-Goldstone mass param-eter are efficiently controlled by the joint effect of super-symmetry and of the global symmetry, allowing in princi-ple the electroweak symmetry to be broken with little fine-tuning, the specific model studied in this paper fails to bringthe Higgs mass above the LEP bound due to a suppressedtree-level quartic coupling. To cure this problem, we con-sider the possibility of additional SU(3)-breaking contribu-tions to the Higgs potential, and estimate the resulting fine-tuning.

The paper is organized as follows. In Sect. 2, we de-scribe the model and discuss the generation of the soft super-symmetry breaking terms via gauge mediation. In Sect. 3,we study the spontaneous symmetry breaking of the globalSU(3) symmetry and find a region of the parameter spacein which the desired vacuum is indeed the global minimumof the scalar potential. We then show how the μ/Bμ prob-lem is solved by the global symmetry. Section 4 deals withelectroweak symmetry breaking, the Higgs mass and fine-tuning. Finally, we give our conclusions in Sect. 5.

2 The model

The model we study in this letter is based on a supersym-metric version [15] of the “simplest little Higgs model” ofRef. [18]. We describe its general structure below, beforediscussing the generation of the soft supersymmetry break-ing terms via gauge mediation.

In order to realize the Higgs as pseudo-Goldstone bosonidea, a global SU(3) symmetry spontaneously broken at thescale f ∼ 1 TeV is imposed on the Higgs sector. The MSSMHiggs doublets Hd and Hu are extended to global SU(3)

(anti-)triplets:

Hd =(

Hd

Sd

)∈ 3, HT

u =(

iσ 2Hu

Su

)∈ 3, (1)

where Su and Sd are electroweak singlets. Similarly, all mat-ter fields are extended to SU(3) multiplets. The global sym-metry is spontaneously broken by the VEVs of Su and Sd .The associated Goldstone boson, which is identified withthe Standard Model Higgs boson H , is a linear combinationof the SU(2)L doublets Hd and iσ 2H ∗

u . By construction,the tree-level Higgs potential does not contain a mass termfor H . However, the global SU(3) symmetry of the Higgssector is not a symmetry of the full Lagrangian: it is vio-lated explicitly by the Yukawa and gauge interactions of theMSSM. These induce a one-loop potential for H which, dueto the combined effect of supersymmetry and of the approx-imate global SU(3) symmetry, has no logarithmic depen-dence on the ultraviolet cut-off [15]. Thanks to this softeningof the radiative corrections, the LEP Higgs mass bound canbe satisfied with less fine-tuning than in the MSSM.

In order for this double protection mechanism to be op-erative, the gauge symmetry must be compatible with theglobal symmetry in the ultraviolet. To this end, the elec-troweak gauge symmetry SU(2)L × U(1)Y is extended toSU(3)W × U(1)X , where Y = X − T 8/

√3. The breaking

of the extended gauge group is achieved at some higher en-ergy scale F � f by two additional Higgs (anti-)triplets ΦD

and ΦU . In this way, the masses of the heavy gauge bosonsare unrelated to the global symmetry breaking scale f , andexperimental limits on them do not constrain it. It is thenpossible to choose f around the TeV scale, so as to mini-mize the fine-tuning in the Higgs potential, without runninginto conflict with precision electroweak data [15].

The details of the model are presented below.

2.1 The SU(3)-symmetric Higgs sector

The Higgs sector has a global SU(3)1 × SU(3)2 symmetrywhose diagonal subgroup is the SU(3)W gauge symmetry. Itcontains the following Higgs multiplets:

– ΦD and ΦU , transforming as 3 and 3 of SU(3)1,

Eur. Phys. J. C (2011) 71:1811 Page 3 of 14

– Hd and Hu, transforming as 3 and 3 of SU(3)2,– two SU(3)1 × SU(3)2 singlets N and N ′.Under SU(3)W × U(1)X , ΦD and Hd (ΦU and Hu) havequantum numbers 3−1/3 (3+1/3), while N and N ′ are sin-glets. The MSSM Higgs doublets Hd and Hu are embeddedin Hd and Hu as indicated in (1). The SU(3)1 × SU(3)2

symmetric Higgs superpotential is chosen to be

WHiggs = λ′N ′(

ΦUΦD − F 2

2

)+ λN HuHd + κ

3N3, (2)

where the last two terms are reminiscent of the NMSSM[19–21]. The role of the N HuHd coupling is to induce thebreaking of the global SU(3)2 symmetry once the singletfield N acquires a VEV. This coupling is also responsible,as in the NMSSM, for the generation of the μ and Bμ pa-rameter through the VEVs of the scalar and F-term compo-nents of N . The last term is crucial to avoid a runaway ofthe tree-level scalar potential in the N direction.

The superpotential (2) leads to the spontaneous break-ing of the global SU(3)1 symmetry, together with the gaugesymmetry breaking SU(3)W × U(1)X → SU(2)L × U(1)Y :

〈ΦD〉 =⎛⎝ 0

0FD

⎞⎠ , 〈ΦU 〉 = (

0 0 FU

), (3)

with FU = FD = F/√

2 in the supersymmetric limit (softterms will shift the VEVs of ΦD and ΦU by an amountO(m2

soft/F )). The spontaneous breaking of the global SU(3)2

symmetry is triggered by a tadpole term in the singlet fieldN , whose origin will be discussed in Sect. 2.3:

〈Hd〉 =⎛⎝ 0

0f cosβ

⎞⎠ , 〈Hu〉 = (

0 0 f sinβ), (4)

where we have defined1 tanβ ≡ 〈Su〉/〈Sd〉.One can take advantage of the hierarchy F � f to

integrate out at the scale F the heavy components ofthe chiral superfields ΦD , ΦU and N ′, as well as theheavy gauge supermultiplets living in the coset SU(3)W ×U(1)X/SU(2)L × U(1)Y . The resulting effective field the-ory is then used to study the breaking of the global SU(3)2

symmetry and of the electroweak symmetry.

2.2 The top quark sector

Like the Higgs fields, the matter fields of the MSSM must beextended to SU(3) multiplets. Since we are mostly interested

1This notation, which is reminiscent of the one used in the MSSM forthe ratio of the two Higgs doublet VEVs, is motivated by the fact thatthe pseudo-Goldstone boson is given by the same linear combinationH � cosβHd + sinβ(iσ 2H ∗

u ) as the lightest MSSM Higgs boson inthe decoupling regime [22, 23].

in electroweak symmetry breaking, we only need to considerthe top/stop sector, which gives the dominant contribution tothe one-loop effective potential.2 For definiteness, we makethe same choice as Ref. [15] for the representations of thetop quark superfields and for their couplings to the Higgssuperfields (see e.g. Refs. [24, 25] for alternative choices):

Wtop = y1ΦUΨQT c + y2 HuΨQtc, (5)

where ΨQ = (QT ,T )T = ((t, b), T )T is an SU(3)W triplet,while tc and T c are SU(3)W singlets (obviously, a sec-ond singlet is necessary to render both the top quark andits SU(3)W partner T massive). Below the scale F , thefirst coupling in (5) is replaced by the effective mass termy1FT T c. The simultaneous presence of the two terms vi-olates the global SU(3)2 symmetry; hence all SU(3)2-breaking effects from the top/stop sector will be proportionalto y1y2.

2.3 Soft supersymmetry breaking terms

So far the model described above is identical to the oneof Ref. [15]. The difference lies in the soft supersymme-try breaking terms, which in our model are calculable interms of a few parameters. Namely, we assume that su-persymmetry is broken in a secluded sector and communi-cated to the observable sector via gauge interactions. As iscustomary, we parameterize supersymmetry breaking by agauge-singlet spurion superfield X and couple it to a vector-like pair of chiral messenger superfields (Φ , Φ), which wechoose to be in the representation (3,1, 1

3 ) ⊕ (1,3,− 13 ) of

SU(3)C × SU(3)W × U(1)X and its conjugate. In order togenerate soft terms for the singlet superfield N , we also in-troduce a coupling NΦΦ:

Wmess = XΦΦ + ξNΦΦ, 〈X〉 = M + θ2FX. (6)

The soft supersymmetry breaking terms for the gauginos andgauge non-singlet scalars are generated by the standard mes-senger loops [1–7], and are schematically given by (the ex-plicit formulas can be found in Appendix B):

mgaugino ∼ α

4πΛ, m2

scalar ∼(

α

4π

)2

Λ2, Λ ≡ FX

M.

(7)

These expressions are valid at the messenger scale M , withα = g2(M)/4π , where g(μ) is the relevant running gauge

2As we are going to see, tanβ turns out to be small in this model, sothat the bottom/sbottom contribution to the one-loop effective potentialcan be neglected.

Page 4 of 14 Eur. Phys. J. C (2011) 71:1811

coupling. The A-terms associated with the Yukawa cou-plings of the top sector, Ay1 and Ay2 , vanish at the mes-senger scale and are generated at lower scales by renormal-ization group running. Due to the direct coupling betweenN and the messenger fields, soft terms for the gauge-singletsuperfield N are also generated (by contrast, the soft termsfor the singlet N ′ vanish). Using the wave-function renor-malization technique of Ref. [26], we find

Aλ = Aκ

3= − 6ξ2

16π2Λ, (8)

m2N = 1

(16π2)2

(48ξ4 − 24κ2ξ2 − 16g2

Cξ2

−16g2Wξ2 − 8

3g2

Xξ2)

Λ2, (9)

where gC , gW and gX are the SU(3)C , SU(3)W and U(1)X

gauge couplings, respectively. Note that m2N < 0 as soon as

ξ � gW . A negative contribution to the soft masses of theHiggs triplets Hd and Hu is also induced by the ξ coupling,on top of the standard (positive) gauge mediation contribu-tion:

m2u = m2

d = 1

(16π2)2

(8

3g4

W + 8

27g4

X − 6λ2ξ2)

Λ2. (10)

Last but not least, the presence of a direct coupling be-tween the singlet N and the messenger superfields also in-duces a tadpole in the scalar potential [8, 27, 28]:

Vtad = m3N + h.c., m3 = 6ξ

16π2Λ2M, (11)

which plays an essential role in the breaking of the globalSU(3)2 symmetry, as well as an effective tadpole term in thesuperpotential [27, 28]:

Wtad = M2NN, M2

N ∼ 6ξ

16π2F ∗

X. (12)

Let us note in passing that we could have avoided thegeneration of a tadpole for N by introducing a second pairof messenger fields with the following superpotential cou-plings [26, 29]:

Wmess = XΦ1Φ1 + XΦ2Φ2 + ξNΦ1Φ2. (13)

Since X and N couple to different combinations ΦiΦj , notadpole arises at one loop and the breaking of the globalSU(3)2 symmetry is triggered by the singlet soft terms (withm2

N < 0 for ξ � gW ). In this letter, we choose to stick tothe minimal case involving a single pair of messenger fields,with the superpotential (6).

3 Spontaneous breaking of the global SU(3)2 symmetry

In order to proceed with the analysis of the global SU(3)2

symmetry breaking, it is convenient to integrate out thefields that acquire a mass of order F when the gauge sym-metry SU(3)W × U(1)X breaks down to SU(2)L × U(1)Y .One is then left with the following tree-level potential forthe fields N , Hd and Hu, valid for energy scales E � F :

VHiggs = VF + VD + Vsoft + δVsoft + Vtad, (14)

VF = ∣∣λHuHd + κN2 + M2N

∣∣2

+ |λ|2|N |2(|Hu|2 + |Hd |2), (15)

VD = g2

8

∑i

(H †

uσ iHu + H†d σ iHd

)2

+ g′2

8

(|Hu|2 − |Hd |2)2, (16)

Vsoft = m2N |N |2 + m2

u|Hu|2 + m2d |Hd |2

+(

λAλN HuHd + κ

3AκN3 + h.c.

), (17)

δVsoft = (m2

D − m2U

)[1

2

(S∗

uSu − S∗dSd

)

+ 9 − 21t2W + 4t4

W

36

(H †

uHu − H†d Hd

)], (18)

Vtad = m3N + h.c., (19)

where g and g′ are the SU(2)L and U(1)Y gauge couplings3,tW ≡ g′/g and the expressions for the soft terms m2

N , m2u,

m2d , Aλ, Aκ and tadpole parameters m3, M2

N have beengiven before. VD contains the SU(2)L × U(1)Y D-termsand breaks the global SU(3)2 symmetry. The term δVsoft,which is a residue of integrating out the heavy gauge super-multiplets [30, 31], also breaks SU(3)2; it is proportional tom2

D − m2U , the difference between the soft masses squared

of the SU(3)1 Higgs triplets ΦD and ΦU [15, 16]. This termis potentially dangerous because it gives a tree-level massto the Higgs boson, thus spoiling its pseudo-Goldstone na-ture. However, since ΦD and ΦU are in conjugate represen-tations, they have equal soft masses at the messenger scaleand the splitting m2

D − m2U is generated by the running be-

tween M and F ; hence it is expected to be small. Indeed,numerical calculations show that the effect of δVsoft on thedynamics of SU(3)2 breaking and on the value of the Higgsmass is negligible.

3The matching conditions between the SU(3)W × U(1)X andSU(2)L × U(1)Y gauge couplings at the scale F read g = gW and

g′ = gW gX/

√g2

W + g2X/3.

Eur. Phys. J. C (2011) 71:1811 Page 5 of 14

The tadpole (19) triggers a VEV vN ≡ 〈N〉 ∼ m ∼(6ξMm2

soft/α2)1/3, together with f ∼ m. On the other hand,

f � 1 TeV is needed in order not to spoil the pseudo-Goldstone nature of the Higgs boson. This points toward arather small value of ξ , namely ξ � 10−3α2(1 TeV/msoft)

2

×(100 TeV/M). As a result the soft terms Aλ, Aκ andm2

N , as well as the negative contribution to m2u and m2

d , arestrongly suppressed and can be neglected in the minimiza-tion of the scalar potential. As for the superpotential tad-pole term, which is of order M2

N ∼ (6ξ/4π)Mmsoft/α, itsonly effect is to shift vN by a relative amount M2

N/m2 ∼(α/4π)m/msoft � (α/4π)(1 TeV/msoft), and we will omitit in the following analytical considerations. Neverthelessall these parameters are included in our numerical compu-tations.

While the VEV of N is stabilized by the superpotentialterm κN3/3, there is no term in V to stabilize a VEV of Su

(Sd ) triggered by a negative m2u (m2

d ). Since m2u is driven

negative by the renormalization group running between themessenger scale M � 100 GeV and the gauge symmetrybreaking scale F , this leads to a runaway in the direction〈N〉 = 〈Sd〉 = 0, |〈Su〉| → ∞, which we discuss in the nextsubsection.

3.1 Global SU(3)2 symmetry breaking: analyticaldiscussion

Let us first ignore the runaway direction and minimize thescalar potential for vN ≡ 〈N〉 �= 0. We have argued in theprevious subsection that the soft terms associated with N aswell as the superpotential tadpole parameter M2

N can be ne-glected to a good approximation. Furthermore, for reasonsthat will become clear later, a tadpole parameter m some-what larger than the scale of MSSM soft terms is needed inorder for the proper symmetry breaking vacuum to be theglobal minimum of the scalar potential. This leads to theprediction of a small tanβ , since the minimization condi-tions give

tan2 β = λ2v2N + m2

d

λ2v2N + m2

u

, (20)

together with vN ∼ m. Neglecting all soft terms (includingthe Higgs soft masses mu and md ) in the minimization of thescalar potential, we obtain the following approximate solu-tion:

v3N = m3

2λ(2κ + λ), f = ± [−2(κ + λ)]1/2

λ5/6[2(2κ + λ)]1/3m, (21)

together with tanβ = 1, for κ + λ < 0. We thus see that thespontaneous breaking of the global SU(3)2 symmetry is in-duced by the tadpole term. To ensure that this occurs not toofar above the electroweak scale, while m can be in the multi-TeV range, some tuning between κ and λ is needed. If one

quantifies the level of tuning by the parameter ε > 0, whereκ = −λ(1 + ε), then f � ∓√

εm/λ2/3. Also,

f

vN

= ±√2ε, (22)

implying vN > f .The problem of the runaway in the Su direction remains

to be discussed. As we show now, this direction is upliftedby radiative corrections. The dominant contribution to theColeman–Weinberg one-loop effective potential for Su,

�V1−loop(Su) = 1

64π2STr

[M4(Su)

(ln

M2(Su)

Λ2− 3

2

)],

(23)

comes from the (s)top sector. The fermion mass matrix isgiven by

(t T

)(0 0

y2Su y1F

)(tc

T c

)(24)

(where we have frozen 〈Hu〉 = 0), and has a single nonzeroeigenvalue corresponding to the mass of the heavy topquark, mT = √|y2Su|2 + |y1F |2. Neglecting the small dif-ference between the soft masses of the stop fields (see Ap-pendix B), the (s)top sector contribution to the one-loop ef-fective potential reads

�V1−loop = 3

16π2

[−m4

T

(ln

(m2

T

Λ2

)− 3

2

)

+ (m2

T + m2stop

)2(

ln

(m2

T + m2stop

Λ2

)− 3

2

)],

(25)

where m2stop ≡ m2

ΨQ= m2

T c = m2tc . For large Su values,

�V1−loop(Su) grows as |Su|2 ln |Su|2, thus curing the run-away behavior due to the m2

u|Su|2 term in the tree-level po-tential. However, an unwanted minimum appears along theSu direction at the location:

S2u,min � F 2

y22

exp

(− 8π2m2

u

3y22m2

stop

), N � Sd � 0, (26)

where we have set Λ = F and, consistently, m2u stands for

the running mass squared m2u(F ). The value of Su,min grows

exponentially as the absolute value of m2u increases. The

value of the scalar potential at this minimum thus decreasesexponentially with |m2

u|:

Vrun(Su,min) � −3F 2m2stop

8π2exp

(− 8π2m2

u

3y22m2

stop

), (27)

where Vrun(Su) = m2u|Su|2 + �V1−loop(Su). In order to

ensure that the global minimum of the scalar potential

Page 6 of 14 Eur. Phys. J. C (2011) 71:1811

is the one approximated by (21), one has to check thatVmin(f, vN) < Vrun(Su,min), where Vmin(f, vN) is the valueof the scalar potential at the minimum (21):

Vmin(f, vN) � 3

2m3vN � − 3m4

2|2λ(2κ + λ)|1/3. (28)

This requirement imposes some restrictions on the parame-ters of the model, most notably on the messenger mass M

and on the tadpole scale m (or equivalently on M and on theparameter ξ ). In particular, larger values of m are preferred,making it necessary to slightly tune the values of κ and λ inorder to maintain the SU(3)2 breaking scale f below 1 TeVor so. Numerical calculations show that M � 1000 TeV andξ � 10−4 lead to reasonable results.

3.2 Global SU(3)2 symmetry breaking: numerical results

In order to study numerically the spontaneous breaking ofthe global SU(3)2 symmetry, we first set the values of theparameters F , y1, y2, κ , λ and of the various soft terms atthe messenger scale and perform the appropriate renormal-ization group (RG) running. The SU(3)W breaking scale F

and the coupling y1 cannot be chosen too large if one wantsto uphold the pseudo-Goldstone nature of the Higgs dou-blet, since the SU(3)2 violating effects in the (s)top sectorare proportional to y1F , as discussed in the next section. Allthe effects neglected in the analytical discussion above (likethe superpotential tadpole term, the soft terms in the Higgspotential and the contributions from the Higgs sector to theColeman–Weinberg effective potential) are taken into ac-count numerically. The SU(3)2-symmetric RGEs valid be-tween the messenger scale M and the SU(3)W breakingscale F can be found in Appendix A, while Appendix Bgives the boundary conditions for the soft terms at the mes-senger scale, calculated using the wave-function renormal-ization technique of Ref. [26].

The minimization of the scalar potential at the scale F

leads to the spontaneous breaking of the SU(3)W × U(1)Xgauge symmetry, together with the SU(3)1 global symme-try. Then the heavy degrees of freedom are integrated outand the breaking of the global SU(3)2 symmetry is studiedby minimizing the effective potential below F , defined asthe sum of the tree-level potential (14) with its parametersrenormalized at the scale F and of the Coleman–Weinbergone-loop corrections computed with Λ = F . The result isthen confronted with the requirement of having the properglobal minimum of the potential, with the SU(3)2 break-ing scale f not too far above the electroweak scale, and acorrect prediction for the top quark Yukawa coupling. Theapproximate proportionality between the VEVs vN , f andthe tadpole scale m, (21), is confirmed by numerical calcu-lations. A slightly more accurate set of equations for vN and

Fig. 1 f [TeV] as a function of λ and ε, for M = 500 TeV,msoft ∼ 1 TeV, y1 = 0.1, F = 10 TeV and ξ = 0.002

Fig. 2 f [TeV] as a function of λ and ε, for M = 500 TeV,msoft ∼ 1 TeV, y1 = 1, F = 7 TeV and ξ = 0.00014

f is given by

f 2 � 1

λ2

[−2λ(κ + λ)v2N − m2

u − m2d

], (29)

2λ2(2κ + λ)v3N + (κ + λ)

(m2

u + m2d

)vN − λm3 � 0, (30)

and the prediction tanβ � 1 holds. Figures 1 and 2 illustratethe dependence of the SU(3)2 breaking scale f on the pa-rameters κ and λ, for a messenger mass M = 500 TeV, atypical soft mass scale msoft ∼ 1 TeV and various choicesfor ξ , y1 and F . All coupling values in the figures are givenat the messenger scale. As shown by these plots, f � 1 TeVcan be achieved with λ ∼ 1 and a mild tuning between κ

and λ, ε ∼ 0.2. These values imply a moderate hierarchybetween the SU(3)2 breaking scale f and the VEV vN .

3.3 The solution of the μ/Bμ problem

Let us now discuss the generation of the μ and Bμ terms.After spontaneous breaking of the global SU(3)2 symmetry,the doublet and singlet components of the Higgs triplets Hu

Eur. Phys. J. C (2011) 71:1811 Page 7 of 14

and Hd no longer share the same masses and couplings, andthe quadratic part of the tree-level scalar potential (14) canbe rewritten as

Vquadr. = |μ|2(|Hu|2 + |Hd |2) + (BμHu · Hd + h.c.)

+ 1

2(su sd sN)M2

S

⎛⎝ su

sdsN

⎞⎠

+ 1

2(pu pd pN)M2

P

⎛⎝ pu

pd

pN

⎞⎠ , (31)

where Hu · Hd ≡ HTu iσ 2Hd , Su,d ≡ fu,d + (su,d +

ipu,d)/√

2, N ≡ vN + (sN + ipN)/√

2 and all parametersin the scalar potential are assumed to be real. As in theNMSSM, the μ and Bμ parameters are generated by theVEVs of the scalar and F-term components of the singletsuperfield N (the signs in the expressions for μ and Bμ aredue to the fact that HuHd = −HT

u iσ 2Hd + SuSd ):

μ = −λvN,

Bμ = λFN − λAλvN (32)

= −λ(λfufd + κv2

N + AλvN

).

However, there is a crucial difference with the NMSSM:here FN receives a contribution from the VEVs of Su andSd , which transform non-trivially under the global SU(3)2

symmetry. At the minimum of the tree-level scalar poten-tial (14), these VEVs take values such that the relation(Bμ)2 = (μ2 + m2

u)(μ2 + m2

d) is satisfied. This in turnsimplies that the determinant of the (Hu, Hd ) mass matrixvanishes (a similar mechanism is at work in the SU(3)-symmetric version of the model of Ref. [8]). The masslesscombination:

H = sinβ(iσ 2H ∗

u

) + cosβHd, (33)

to be identified with the Standard Model Higgs boson, is in-terpreted as a Goldstone boson of the spontaneously brokenglobal SU(3)2 symmetry. The orthogonal combination H ′ isheavy with a mass m2

H ′ = 2μ2 +m2u +m2

d . Inspection of thesinglet scalar and pseudoscalar mass matrices M2

S and M2P

show that there is another Goldstone boson,

η = sinβpu − cosβpd. (34)

The other singlets are massive with masses of order a few μ,except for a lighter one with a mass of order λf � 1 TeV.

If we restrict our attention to the part of the tree-levelscalar potential that depends solely on H , we see no depen-dence on μ and Bμ, while the masses of the heavy statesof the Higgs sector (including the higgsinos) are of orderμ. Hence, the electroweak scale is insensitive to the actual

value of the μ parameter, which is allowed to be large with-out creating a strong fine-tuning in the Higgs potential4. Thiselegantly solves the μ/Bμ problem of gauge mediation. Thevalue of μ (or vN ) is relevant, on the other hand, for thebreaking of the global symmetry, and we have seen in theprevious subsections that a moderate hierarchy μ � f isneeded, with no incidence on fine-tuning by virtue of theglobal SU(3)2 symmetry.

4 Electroweak symmetry breaking

We are now in a position to discuss the breaking of the elec-troweak symmetry. Let us first recapitulate the identifica-tion of the light degrees of freedom in the Higgs sector. Thespontaneous breaking of the global SU(3)1 × SU(3)2 sym-metry leads to 10 Goldstone bosons, 5 of which disappearfrom the massless spectrum by virtue of the Higgs mecha-nism, since the gauge symmetry SU(3)W ×U(1)X is brokento SU(2)L × U(1)Y in the same process. The remaining 5Goldstone bosons reside mainly in Hu and Hd in the limitf � F , and are conveniently parameterized as5 [15]:

Hd = fd

⎛⎜⎝

H|H | sin

( |H |f

)

e− iη

f√

2 cos( |H |

f

)⎞⎟⎠ ,

Hu = fu

(H †

|H | sin

( |H |f

), e

iη

f√

2 cos

( |H |f

)),

(35)

where fu ≡ f sinβ , fd ≡ f cosβ and |H | ≡ √H †H . All

other components of the Higgs triplets, except for one sin-glet with mass of order λf , are heavy with masses of or-der a few μ � f and can be integrated out. H is a Stan-dard Model-like Higgs doublet transforming as a 2−1/2 ofSU(2)L × U(1)Y , while η is a singlet whose role has beendiscussed in Refs. [15, 24, 25]. Being a (pseudo-)Goldstoneboson of the approximate global SU(3)2 symmetry, H getsits potential from SU(3)2 breaking interactions. At treelevel, one has

Vtree(H) = Vlight(H) + Vheavy(H), (36)

where Vlight(H) is the contribution of the SU(2)L × U(1)YD-terms:

Vlight(H) = λ0

{|H |4 + O

( |H |6f 2

)},

λ0 = g2 + g′2

8cos2 2β,

(37)

4In fact, radiative corrections induce a dependence of the one-loop ef-fective potential on μ and Bμ (see Sect. 4.1), but this does not repre-sent an important source of fine-tuning.5This parametrization agrees with (33) and (34) at leading order in 1/f .

Page 8 of 14 Eur. Phys. J. C (2011) 71:1811

and Vheavy(H) is the contribution of the terms δVsoft leftover by integrating out the heavy gauge supermultiplets atthe scale F :

Vheavy(H) = m20

{|H |2 + O

( |H |4f 2

)},

m20 = 9 − 21t2

W + 4t4W

36

(m2

D − m2U

)(− cos 2β)

(38)

(m20 > 0 due to cos 2β < 0 and m2

D − m2U > 0). Since

tanβ � 1, both the tree-level quartic coupling λ0 and themass parameter m2

0 are small.6 As we are going to see inthe next subsection, one-loop corrections induced by thelarge top quark Yukawa coupling generate a tachyonic massterm in the Higgs potential and trigger electroweak symme-try breaking. The electroweak scale v is related to the VEVof the Higgs doublet v ≡ 〈H 〉 by

v = f sin(v/f ). (39)

4.1 Electroweak symmetry breaking: analytical discussion

At the one-loop level, V (H) receives contributions fromthe Higgs couplings to the matter and gauge fields, whichexplicitly break the global SU(3)2 symmetry. Let us firstcompute the radiative corrections induced by the top quarkYukawa coupling, using the Coleman–Weinberg formula((23) with Su replaced by H ). The fermion mass matrixsquared reads, in the parameterization (35):

M†topMtop

=(

y22f 2

u y1y2Ffu cos(|H |/f )

y1y2Ffu cos(|H |/f ) y21F 2

), (40)

where Lmass � −(t T )Mtop(tc T c)T + h.c. The eigenstates

can be identified with the Standard Model top quark and itsheavy SU(3) partner, with masses:

(mT

t

)2 = 1

2

(y2

1F 2 + y22f 2

u

±√(

y21F 2 + y2

2f 2u

)2 − 4y21y2

2F 2f 2u sin2(|H |/f ))

.(41)

For |H | � f , (41) simplifies to

m2t � y2

t |H |2, m2T � y2

1F 2 + y22f 2

u , (42)

6As discussed at the beginning of Sect. 3, m20 is further suppressed

by the small RG-induced difference of heavy Higgs triplet soft massesm2

D − m2U . It is therefore not expected to play a significant role in the

dynamics of electroweak symmetry breaking. This is confirmed by nu-merical calculations, which show that the tree-level contributions to theHiggs potential are negligible in comparison with the one-loop correc-tions.

where

y2t = y2

1y22F 2 sin2 β

y21F 2 + y2

2f 2u

. (43)

Plugging these expressions into the Coleman–Weinberg for-mula and neglecting the small difference between the softmasses of the stop fields, one obtains similar expressions tothe ones of Ref. [15]:

δtm2H � − 3y2

t

8π2

[m2

stop ln

(1 + m2

T

m2stop

)

+ m2T ln

(1 + m2

stop

m2T

)], (44)

and

δtλH � 3y4t

16π2

[ln

(m2

stopm2T

m2t (m

2stop + m2

T )

)

− 2m2

stop

m2T

ln

(1 + m2

T

m2stop

)

+ 2m2stop

3y2t f 2

ln

(m2

stop + m2T

m2stop

)

+ 2m2T

3y2t f 2

ln

(m2

stop + m2T

m2T

)], (45)

where δtm2H and δtλH are the contributions of the (s)top

sector to the coefficients of the quadratic and quartic termsin the one-loop effective Higgs potential, respectively:

�V1−loop(H) = δm2H |H |2 + δλH |H |4 + · · · . (46)

As required for proper electroweak symmetry breaking,δtm

2H is negative while δtλH is positive. The absence of a

lnΛ-dependent piece in δtm2H is a direct consequence of the

double protection of the Higgs mass by supersymmetry andby the global SU(3)2 symmetry.

Let us now consider the contributions of the gauge inter-actions to δm2

H and δλH . In the effective theory below thegauge symmetry breaking scale F , these depend logarithmi-cally on the cut-off scale Λ due to the explicit breaking ofthe global SU(3)2 symmetry by the SU(2)L × U(1)Y gaugecouplings and gaugino masses. The dominant contributionsto the one-loop effective potential are given by the approxi-mate formula:

�gaugeV1−loop(H)

� − 1

64π2STr

[M4

gauge−Higgs(H) lnΛ2

m2soft

], (47)

where M4gauge−Higgs stands for the (fourth power of the)

mass matrices of the gauge and Higgs fields, and msoft

Eur. Phys. J. C (2011) 71:1811 Page 9 of 14

is an average soft mass. Since the whole gauge sector isSU(3)2 symmetric above the scale F , the logarithmic diver-gence is effectively cut off at Λ = F . Let us compute (47).Working in the approximation where the heavy gauge andHiggs fields are integrated out at tree level, one is left withthe SU(2)L × U(1)Y gauge fields and with the Higgs su-perfields Hu, Hd and N . Using the Higgs superpotentialW = λN HuHd + κ

3 N3, the tree-level potential (14)–(19)and the Lagrangian terms involving the SU(2)L × U(1)Ygauge fields, one derives the mass matrices of the gaugebosons, charginos, neutralinos, charged and neutral Higgsbosons. Neglecting δVsoft as well as the soft terms that aresuppressed by the small parameter ξ , and assuming all pa-rameters to be real, one obtains

STr[

M4gauge−Higgs

] = −[4(3g2M2

2 + g′2M21

)+ 3

(3g2 + g′2)μ2](|Hu|2 + |Hd |2)

+ ([4(3g2M2 + g′2M1

)μ

− (3g2 + g′2)Bμ

]Hu · Hd + h.c.

)+ 3

(g2 + g′2)[m2

u|Hu|2 + m2d |Hd |2]

− 2g′2[m2d |Hu|2 + m2

u|Hd |2]

+[(g2 + g′2)2

2− 9

4g4

+ 1

4g′4 − λ2(g2 + g′2)]

× (|Hu|2 − |Hd |2)2, (48)

where field-independent terms have been omitted, and μ =−λvN , Bμ = −λ(λfufd + κv2

N). Inserting the parameteri-zation (35) into (48), one finally obtains

δgm2H �

{3g2M2

2 + g′2M21

16π2+ 3

64π2

(3g2 + g′2)μ2

+ 3g2M2 + g′2M1

16π2μ sin 2β

− 3g2 + g′2

64π2Bμ sin 2β

− 3(g2 + g′2)64π2

[m2

u sin2 β + m2d cos2 β

]

+ g′2

32π2

[m2

d sin2 β + m2u cos2 β

]}ln

(F 2

m2soft

),(49)

δgλH � − 1

64π2

[(g2 + g′2)2

2− 9

4g4 + 1

4g′4

− λ2(g2 + g′2)] cos2 2β ln

(F 2

m2soft

)− δgm

2H

3f 2,

(50)

where we have set Λ = F . The terms enhanced by μ2 andμM1,2 dominate in δgm

2H , so that δgm

2H > 0. Note that there

is a partial cancellation between the second and the fourthterms, due to Bμ � −λκv2

N � λ2v2N = μ2, leaving a net

contribution (3g2 + g′2)μ2 ln(F 2/m2soft)/32π2. Contrary to

δtλH , δgλH is negative, but it is suppressed by cos2 2β (firstterm) and by 1/3f 2 (second term).

The second and fourth terms in (49), which due to thelarge value of μ (see next subsection) give the dominantcontribution from the gauge-Higgs sector to the Coleman–Weinberg potential, have a simple renormalization group in-terpretation. They arise from the different RG running, be-low the SU(3)W breaking scale F , of the parameters asso-ciated with the doublet and singlet components of the Higgstriplets Hu and Hd . Indeed, below F , gauge interactions dis-tinguish the doublets Hu and Hd from their SU(3) partnersSu and Sd , and this results in different RGEs for parametersthat would otherwise be equal by virtue of the SU(3)2 sym-metry. One is thus led to “split” the superpotential couplingλ in the following way:

WHiggs � λsNSuSd + λdNHuHd, (51)

and similarly for the soft terms involving Hu or Hd . As aresult, the F -term potential (15) is modified as follows:

VF = v2Nf 2

(λ2

s cos2( |H |

f

)+ λ2

d sin2( |H |

f

))

+([

λs cos2( |H |

f

)+ λd sin2

( |H |f

)]

× f 2 sinβ cosβ + κv2N

)2

, (52)

where we have replaced N by its VEV and inserted the pa-rameterization (35). Using Bμ � μ2 and sin 2β � 1, thisyields:

δsplitm2H � λd − λs

λμ2, δsplitλH � −δsplitm

2H

3f 2, (53)

where λd − λs/λ can be computed from the RGEs for the“split” superpotential couplings given in Appendix C:

λd − λs

λ� ln

λd

λs

� 3g2 + g′2

32π2ln

(F 2

m2soft

), (54)

in agreement with the second and fourth terms of (49).

4.2 Electroweak symmetry breaking: numerical results

The numerical study of electroweak symmetry breaking isdone by minimizing the Higgs potential, taking into ac-count the contributions mentioned in the previous subsec-tions and relaxing the assumptions made in the analytical

Page 10 of 14 Eur. Phys. J. C (2011) 71:1811

discussion (in particular, the soft masses in the stop sectorare not universal but given by the formulas of Appendix B).For simplicity, only the dominant (s)top sector contributionand the approximate gauge contribution (47)–(48) have beenincluded in the numerical computation of the Coleman–Weiberg potential.

Let us comment on the features of the main contribu-tions to the Higgs potential, as revealed by the numericalcalculations. The contribution from the (s)top sector to theColeman–Weinberg one-loop effective potential has the de-sired “mexican hat” shape with the minimum approximatelylocated at v ≈ π

2 f . The gauge contribution is convex forsmall values of the Higgs VEV with a minimum at the ori-gin, which helps shifting the minimum of the Higgs po-tential toward the correct value v ≈ v = 174 GeV. How-ever, the LEP bound on the Higgs mass requires large cor-rections from the (s)top sector to the quartic coupling λH

and at the same time does not allow for large correctionsfrom the gauge-Higgs sector (which gives δgλH < 0). Thisin turn implies that the gauge contribution (47)–(48) is notenough to bring v to its true value. Conversely, one may ad-just the parameters of the model so that the correct valueof the electroweak scale is obtained, but then the radiativecorrections to λH are too small and the Higgs mass falls be-low the LEP bound. This means that the model must be ex-tended to be fully realistic. A first way to do so is to addan SU(3)2-breaking sector that generates a sizeable tree-level quartic coupling λ0, as was done in a different SU(3)

model in Ref. [16]. Then large corrections from the (s)topsector are no longer needed to satisfy the LEP bound, andthe electroweak scale is obtained with little fine-tuning. An-other possibility is to invoke some additional convex con-tribution δextram

2H |H |2 to the effective Higgs potential (pre-

sumably arising from loops involving the heavy gauge andHiggs fields, or from some extra SU(3)2-breaking sector tobe added to the model) in order to obtain the proper valueof the Higgs VEV. In this case large corrections from the(s)top sector to λH are still needed to satisfy the LEP con-straint and the fine-tuning is more significant.

Let us investigate the second possibility. To fix the sizeof the (s)top and gauge contributions, we require that the re-sulting δλH be large enough to satisfy the LEP bound for amass of the SU(3) top partner mT in the 1–10 TeV range.In practice, the parameter values chosen in Fig. 2 for thespontaneous breaking of the global SU(3)2 symmetry turnout to be convenient for that purpose and we adopt them inour numerical study. Then we adjust the extra contributionδextram

2H |H |2 to obtain the correct value of the electroweak

scale. Let us now present the numerical results for the samechoice of parameters as in Fig. 2 (all coupling values in thefigures are given at the messenger scale). The value of theHiggs mass Mh � 2

√λ0 + δλH v is displayed in Fig. 3 (the

dashed lines show the Higgs mass predicted by the (s)top

Fig. 3 Higgs boson mass [in TeV] as a function of λ and ε, forM = 500 TeV, msoft ∼ 1 TeV, y1 = 1, F = 7 TeV and ξ = 0.00014

Fig. 4 tanβ as a function of λ and ε. Other parameters chosen as inFig. 3

sector contribution to the effective potential alone), whilethe corresponding value of tanβ is shown in Fig. 4. Thecurves in Figs. 4 to 6 are dashed in the region of the param-eter space where the Higgs mass lies below the LEP bound.

The source of fine-tuning in the model lies in the largeradiative corrections to λH from the (s)top sector that arerequired in order to satisfy the LEP bound. This in turn im-plies a large and negative δtm

2H that must be compensated

for by δgm2H and δextram

2H so as to obtain the proper value of

the electroweak scale v = 174 GeV. One can estimate thisfine-tuning with the following quantity:

FT =∣∣∣∣ |δtm

2H | − |δm2

H |δm2

H

∣∣∣∣, (55)

where δm2H = δtm

2H +δgm

2H +δextram

2H is the mass squared

parameter in the Higgs potential (46). The numerical resultsfor the fine-tuning parameter FT are presented in Fig. 5.Figures 3 to 5 show that successful electroweak symmetrybreaking with a Higgs boson mass above the LEP boundand a fine-tuning around FT ∼ 20 can be achieved for λ ∼ 1,ε ∼ 0.3 and ξ ∼ 10−4 (corresponding to f ∼ 1 TeV).

Eur. Phys. J. C (2011) 71:1811 Page 11 of 14

Fig. 5 Fine-tuning as a function of λ and ε. Other parameters chosenas in Fig. 3

Fig. 6 μ = −λvN as a function of λ and ε. Other parameters chosenas in Fig. 3

4.3 Physical spectrum

Let us finally discuss the physical spectrum of the model.The gross features of the Higgs spectrum are the follow-ing (omitting the heavy fields ΦU , ΦD and N ′, whichhave masses of order F , out of reach of the LHC). Thespontaneous breaking of the SU(3)2 symmetry yields amassive Higgs doublet H ′ ∼ cosβ(iσ 2H ∗

u ) − sinβHd ,which describes a CP-even and a CP-odd neutral scalarsas well as a charged one, all with the same tree-level mass√

2μ2 + m2u + m2

d ≈ √2|μ|, hence in the multi-TeV range

(see Fig. 6). In the singlet sector, we have four heavy scalarswith masses of order a few μ and a lighter one with a massof order λf � 1 TeV; the remaining singlet η is a pseudo-Goldstone boson and gets a small mass at the one-looplevel [24]. Apart from this singlet, whose phenomenologyhas been studied in Refs. [15, 24, 25], the Higgs sector con-tains a single light state with Standard Model-like proper-ties.

The higgsinos (both doublet and singlets) also have largemasses of order μ. The rest of the superpartner spectrum is

representative of gauge-mediated models with a low mes-senger scale. The SU(3) partner of the top quark has a massmT � y1F ≈ 7 TeV in the region of parameter space consid-ered and is not accessible at the LHC, similarly to the heavygauge bosons associated with the broken SU(3)W × U(1)Xgenerators.

5 Conclusions

In this paper, we studied the interplay between the sponta-neous breaking of a global symmetry of the Higgs sector andgauge-mediated supersymmetry breaking, in the frameworkof a supersymmetric model with global SU(3) symmetry.In addition to solving the supersymmetric flavor problemand alleviating the little hierarchy problem by identifyingthe Higgs boson with a pseudo-Goldstone boson, this sce-nario presents several advantages.

First, gauge mediation provides a mechanism for break-ing the global symmetry protecting the Higgs mass, namelythrough the loop-induced tadpole of an SU(3) singlet scalarfield. A non-trivial success of the model studied in this pa-per, compared with previous attempts in the literature, is toensure that the global symmetry breaking vacuum is indeedthe global minimum of the scalar potential, and the possi-bility to control the shape of the potential by varying thetadpole scale is instrumental in this.

Second, the global symmetry provides an elegant solu-tion to the μ/Bμ problem of gauge mediation. Much likein the NMSSM, the μ and Bμ parameters are generated bythe VEVs of the scalar and F-term components of a singletsuperfield, but the global SU(3) symmetry ensures that therelation (Bμ)2 = (|μ|2 + m2

u)(|μ|2 + m2d) is satisfied at the

minimum of the tree-level scalar potential, implying that theelectroweak scale is insensitive to the actual value of the μ

parameter. As a result the μ parameter, which sets the scaleof the heavy Higgs masses, may be large without creating astrong fine-tuning in the Higgs potential.

Finally, the combined effect of supersymmetry and of theglobal symmetry ensures a “double protection” of the Higgspotential, allowing for a reduced fine-tuning with respect tothe MSSM. We computed the one-loop corrections to thepotential of the pseudo-Goldstone Higgs boson coming fromthe (s)top and gauge-Higgs sectors, and checked that they in-deed trigger electroweak symmetry breaking. However, thespecific model studied in this paper has a suppressed tree-level quartic Higgs coupling and fails to bring the Higgsmass above the LEP bound. We showed that an additionalcontribution δextram

2H |H |2 to the Higgs potential, arising

from some extra SU(3)-breaking sector, can solve this prob-lem with a moderate fine-tuning of order 1/20. Alterna-tively, one may try to generate a sizeable tree-level quarticHiggs coupling along the lines of Ref. [16].

Page 12 of 14 Eur. Phys. J. C (2011) 71:1811

The model predicts a rather low messenger scale, a smalltanβ value, a light Higgs boson with Standard Model-likeproperties, and heavy higgsinos.

Acknowledgements The work of A.K. was supported by the MNi-SzW scientific research grant N N202 103838 (2010–2012). The workof S.L. was partially supported by the European Community under thecontracts MTKD-CT-2005-029466 and PITN-GA-2009-237920.

Appendix A: Renormalization group equations

In this appendix, we give the renormalization group equa-tions (RGEs) valid between the messenger scale M and theSU(3)W breaking scale F for all relevant superpotential pa-rameters and soft terms. For convenience, we recall theirdefinition below:

W � λ′N ′(ΦUΦD − μ2) + λN HuHd + κ

3N3

+ y1ΦUΨQT c + y2 HuΨQtc, (A.1)

Vsoft � M2U |ΦU |2 + M2

D|ΦD|2 + M2N ′ |N ′|2

+ m2u|Hu|2 + m2

d |Hd |2 + m2N |N |2

+ m2ΨQ

|ΨQ|2 + m2tc |mtc |2 + m2

T c |mT c |2

+(

λ′Aλ′N ′ΦUΦd + λAλN HuHd + κ

3AκN3

+ y1Ay1ΦUΨQT c + y2Ay2 HuΨQtc + h.c.

). (A.2)

In the RGEs below, gC , gW and gX are the SU(3)C , SU(3)Wand U(1)X gauge couplings, respectively; MC , MW and MX

are the associated gaugino masses; and t ≡ (1/16π2) lnμ.

d

dtλ′ = λ′

(5λ′2 + 3y2

1 − 16

3g2

W − 4

9g2

X

), (A.3)

d

dtλ = λ

(5λ2 + 2κ2 + 3y2

2 − 16

3g2

W − 4

9g2

X

), (A.4)

d

dtκ = 3κ

(3λ2 + 2κ2), (A.5)

d

dty1 = y1

(7y2

1 + y22 + λ′2 − 16

3g2

C − 16

3g2

W − 4

3g2

X

),

(A.6)

d

dty2 = y2

(y2

1 + 7y22 + λ2 − 16

3g2

C − 16

3g2

W − 4

3g2

X

),

(A.7)

d

dtAλ′ = 10λ′2Aλ′ + 6y2

1Ay1 − 32

3g2

WMW − 8

9g2

XMX,(A.8)

d

dtAλ = 10λ2Aλ + 4κ2Aκ + 6y2

2Ay2

− 32

3g2

WMW − 8

9g2

XMX, (A.9)

d

dtAκ = 6

(3λ2Aλ + 2κ2Aκ

), (A.10)

d

dtAy1 = 14y2

1Ay1 + 2λ′2Aλ′ − 32

3g2

CMC

− 32

3g2

WMW − 8

3g2

XMX, (A.11)

d

dtAy2 = 14y2

2Ay2 + 2λ2Aλ − 32

3g2

CMC

− 32

3g2

WMW − 8

3g2

XMX, (A.12)

d

dtm2

U = 6y21

(m2

U + m2ΨQ

+ m2T c + A2

y1

)

+ 2λ′2(m2U + m2

D + m2N ′ + A2

λ′)

− 32

3g2

WM2W − 8

9g2

XM2X, (A.13)

d

dtm2

D = 2λ′2(m2U + m2

D + m2N ′ + A2

λ′)

− 32

3g2

WM2W − 8

9g2

XM2X, (A.14)

d

dtm2

N ′ = 6λ′2(m2U + m2

D + m2N ′ + A2

λ′), (A.15)

d

dtm2

u = 6y22

(m2

u + m2ΨQ

+ m2tc + A2

y2

)

+ 2λ2(m2u + m2

d + m2N + A2

λ

)

− 32

3g2

WM2W − 8

9g2

XM2X, (A.16)

d

dtm2

d = 2λ2(m2u + m2

d + m2N + A2

λ

)

− 32

3g2

WM2W − 8

9g2

XM2X, (A.17)

d

dtm2

N = 6λ2(m2u + m2

d + m2N + A2

λ

)

+ 4κ2(3m2N + A2

κ

), (A.18)

d

dtm2

ΨQ= 2y2

1

(m2

U + m2ΨQ

+ m2T c + A2

y1

)

+ 2y22

(m2

u + m2ΨQ

+ m2tc + A2

y2

)

− 32

3g2

CM2C − 32

3g2

WM2W − 8

9g2

XM2X, (A.19)

d

dtm2

tc = 6y22

(m2

u + m2ΨQ

+ m2tc + A2

y2

)

− 32

3g2

CM2C − 32

9g2

XM2X, (A.20)

Eur. Phys. J. C (2011) 71:1811 Page 13 of 14

d

dtm2

T c = 6y21

(m2

U + m2ΨQ

+ m2T c + A2

y1

)

− 32

3g2

CM2C − 32

9g2

XM2X. (A.21)

Appendix B: Values of soft terms at the messenger scale

In this appendix, we give the boundary conditions for thesoft terms in (A.2) at the messenger scale M .

m2U = m2

D = 1

(16π2)2

(8

3g4

W + 8

27g4

X

)Λ2, (B.1)

m2u = m2

d = 1

(16π2)2

(8

3g4

W + 8

27g4

X − 6λ2ξ2)

Λ2, (B.2)

m2N ′ = 0, (B.3)

m2N = 1

(16π2)2

(48ξ4 − 24κ2ξ2 − 16g2

Cξ2

− 16g2Wξ2 − 8

3g2

Xξ2)

Λ2, (B.4)

m2ΨQ

= 1

(16π2)2

(8

3g4

C + 8

3g4

W + 8

27g4

X

)Λ2, (B.5)

m2tc = 1

(16π2)2

(8

3g4

C + 34

27g4

X

)Λ2, (B.6)

m2T c = 1

(16π2)2

(8

3g4

C + 34

27g4

X

)Λ2, (B.7)

Aλ = Aκ

3= − 6ξ2

16π2Λ, (B.8)

Aλ′ = Ay1 = Ay2 = 0. (B.9)

Appendix C: Renormalization group equations for“split” superpotential couplings

Below the SU(3)W breaking scale F , the SU(3)2-invariantsuperpotential couplings λ and y2 are split into separate cou-plings for the doublet and singlet components of the Higgstriplets Hu and Hd :

W � λsNSuSd + λdNHuHd

+ y2,sSuT tc + y2,dHuQtc. (C.1)

The associated RGEs, valid below the scale F , are given by

d

dtλs = λs

(3y2

2,s + 3λ2s + 2λ2

d + 2κ2), (C.2)

d

dtλd = λd

(3y2

2,d + λ2s + 4λ2

d + 2κ2

− 2

(1

2g′2 + 3

2g2

)), (C.3)

d

dty2,s = y2,s

(5y2

2,s + 2y22,d + λ2

s

− 2

(8

9g′2 + 8

3g2

C

)), (C.4)

d

dty2,d = y2,d

(y2

2,s + 6y22,d + λ2

d

− 2

(13

18g′2 + 3

2g2 + 8

3g2

C

)), (C.5)

where gC , g and g′ are the SU(3)C , SU(2)L and U(1)Ygauge couplings, respectively, and t ≡ (1/16π2) lnμ. Thematching conditions between the SU(3)W × U(1)X andSU(2)L × U(1)Y gauge couplings at the scale F read

g = gW , g′ = gWgX√g2

W + g2X/3

. (C.6)

For completeness, we also give the matching conditions forthe gaugino masses:

M2 = MW, M1 = g2XMW + 3g2

WMX

3g2W + g2

X

. (C.7)

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