UT-09-22
IPMU 09

Relaxing the constraint on the number of messengers in alow-energy gauge mediation

Ryosuke Sato1,2 , T. T. Yanagida2,1 and Kazuya Yonekura1,2
1

Department of Physics, University of Tokyo,
Tokyo 113-0033, Japan

2

Institute for the Physics and Mathematics of the Universe (IPMU), University ofTokyo,
Chiba 277-8568, Japan

Abstract
We propose a mechanism for relaxing a constraint on the number of messengers in low-energy gauge mediation models. The Landau pole problem for the
standard-model gauge coupling constants in the low-energy gauge mediation can be
circumvented by using our mechanism. An essential ingredient is a large positive
anomalous dimension of messenger ﬁelds given by a large Yukawa coupling in an
conformal ﬁeld theory at high energies. The positive anomalous dimension reduces
the contribution of the messengers to the beta function of the standard-model gauge
couplings.

1

Introduction

The low-energy-scale gauge mediation with the gravitino mass m3/2 < O(10) eV is very
attractive, since it does not suﬀer from any cosmological gravitino problem [1]. In sucha low-energy gauge mediation, the messengers have their masses of the order 102 − 103
TeV. If the number of messengers, Nmess , is large, the gauge coupling constants of thestandard model easily blow up below the GUT scale, i.e. the gauge coupling constants
hit Landau poles below the GUT scale. The requirement of the perturbative uniﬁcationof the gauge coupling constants, thus, leads to a constraint on the number of messengers.
It is known [2] that Nmess < 5 for the messengers being 5 + 5∗ of SU (5)GUT if the massesof messengers are smaller than about 103 TeV.
The above constraint becomes more severe if one considers strongly interacting messengers in direct (see [3] and references therein) or semi-direct [4, 5] gauge mediation
models, for instance. This is because the messengers receive most likely negative anomalous dimensions from the hidden strong gauge interactions and hence the standard-model
gauge couplings run faster (see Section 2).
In this paper we point out that it is not always the case if the theory is embedded into
a conformal ﬁeld theory at high energies. We show several examples where hidden sector
interactions induce even positive large anomalous dimensions for messengers. In thoseexample models one may have Nmess ≥ 5 without ruining the perturbative uniﬁcation.
A crucial ingredient is an introduction of a large Yukawa coupling of the messengers tosome other hidden sector ﬁelds.

2

Relaxing the constraint on Nmess

In this section we describe our mechanism for relaxing the constraint on the number ofmessengers. In supersymmetric (SUSY) gauge theories, the β function of a gauge couplingis exactly given by the so-called NSVZ β function [6],Pg 4 3t(A) − i (1 − γi )t(i)∂,β(g) = µ g 2 = − 2∂µ8π
1 − t(A)g 2 /8π 2(1)where t(A) and t(i) are the Dynkin indices for the adjoint representation and the representation of matter ﬁelds i, γi an anomalous dimension of matter i, and µ a renormalization2

scale. Let us consider the β functions of the standard-model (SM) gauge couplings. Fromthe β function (1), we can see that the eﬀective messenger number contributing to theSM β functions is give by
eﬀ
Nmess
≡

X

(1 − γi )tGUT (i),(2)

i∈messwhere the sum is taken over messenger ﬁelds, and tGUT (i) the Dynkin indices for the GUT
gauge group SU (5)GUT . Here, we have assumed that the messengers form a complete
representation of the SU (5)GUT .
Now let us suppose that the messengers are charged under some hidden gauge group(with the gauge coupling g) and have a Yukawa interaction (with the Yukawa couplingλ) with some hidden matters. Then, the anomalous dimension of the messengers is given
by, at the one-loop level,
γ∼−

g2
|λ|2
+
.
8π 2 8π 2(3)(Here, we have neglected the contributions from the SM gauge interactions.) Then, from
Eqs. (2) and (3), we can see that the hidden gauge interaction increases the eﬀectiveeﬀ
messenger number, Nmess, while the hidden Yukawa interaction decreases it.

In direct (see [3] and references therein) or semi-direct [4, 5] gauge mediation models
and also in composite messenger models (e.g. [7]), the messenger ﬁelds are supposed to be
charged under hidden gauge groups. Thus, the messenger ﬁelds have negative anomalous
dimensions and the eﬀective messenger number increases. Thus, the Landau pole problem
discussed in the Introduction becomes more severe (for a more quantitative discussion, see
Appendix A). However, if we introduce large Yukawa interactions in the messenger sector,
the anomalous dimensions of the messengers can become positive and we can decreaseeﬀ. For this mechanism to be eﬃcient, it is desirable
the eﬀective messenger number, Nmessthat the hidden gauge theory of the messengers is embedded into a conformal ﬁeld theory,
because the messengers can have large positive anomalous dimensions over a wide range
of energy scales. (Otherwise, the large Yukawa coupling hits its own Landau pole below
the GUT scale.)In the rest of this section we give example models which realize the above mechanism.
We will see that the models have direct applications to low-energy gauge mediation models

3

in the next section.
The models are based on an SU (NC ) hidden gauge group. We ﬁrst introduce NQ messenger quarks and anti-quarks, Qiα and Q̃αi , with i = 1, · · · , NQ and α = 1, · · · , NC . The
messengers Qiα and Q̃αi transform as fundamental and anti-fundamental representationsof SU (NC ), respectively. We restrict our discussion to the case of NQ = 5, for simplicity,
and assume that the quarks Qiα and anti-quarks Q̃αi transform as 5∗ and 5 of SU (5)GUT ,respectively. We introduce a mass term mQ Qiα Q̃αi for the messengers Qiα and Q̃αi . Noticethat NC is identiﬁed with the number of the messengers, Nmess . The generalization toother gauge theories such as SP (NC ) or SO(NC ) is straightforward and hence we do notdiscuss it in this paper.
To embed the theory into a conformal ﬁeld theory giving the messengers positive
anomalous dimensions, we introduce NP pairs of quarks and anti-quarks, Pαp and P̃pα with
p = 1, · · · , NP , and an adjoint quark chiral multiplet, Aαβ with α, β = 1, · · · , NC . We
introduce their mass terms,
Wmass = mP Pαp P̃pα + mA Aαβ Aβα .

(4)We assume, mP , mA > mQ , for the additional quarks, Pαp and P̃pα , and the adjoint quarkAαβ to decouple from the strong dynamics at the messenger mass scale. We also introduce
a Yukawa coupling,
WYukawa =√2λQiα Aαβ Q̃βi .

(5)The introduction of the Yukawa coupling is important for our mechanism to work, as
explained above.
We ﬁnd that the theory has a infrared conformal ﬁxed point for a given appropriate
value of NC and that of NP . We show, in Appendix B, the detailed determination of theinfrared ﬁxed points and of the anomalous dimensions of the messenger ﬁelds. We givethe obtained anomalous dimensions γQ of the messenger ﬁelds Q and Q̃ and the eﬀectiveeﬀ= (1 − γQ )NC for various sets of (NC , NP ) in Table 1. We
messenger numbers Nmesseﬀ< 5 for many sets ofsee that the models have the eﬀective messenger numbers Nmess(NC , NP ), even if the actual messenger number Nmess = NC ≥ 5.

4

NC = 5NC = 6NC = 7NC = 8NC = 9NP = 20.303 (3.48)
0.452 (3.29)
×××

NP = 30.156 (4.22)
0.300 (4.20)
0.411 (4.13)
0.494 (4.05)
×

NP = 40.062 (4.69)
0.191 (4.85)
0.301 (4.90)
0.388 (4.89)
0.458 (4.88)

NP = 5×0.110 (5.34)
0.214 (5.50)
0.302 (5.58)
0.375 (5.63)

NP = 6×
0.048 (5.71)
0.144 (5.99)
0.230 (6.16)
0.303 (6.27)

Table 1: The anomalous dimensions γQ of messenger ﬁelds Q, Q̃ and the eﬀective meseﬀ
senger numbers Nmess(in parentheses) at the conformal ﬁxed points. Models marked with× do not have a desirable ﬁxed point.

3

Applications to low-energy gauge mediation models

In this section we discuss applications of our mechanism to various gauge mediation
models. The applications have three categories, (I) application to direct gauge mediation,(II) that to semi-direct gauge mediation and (III) that to composite messenger models. We
consider a representative model for each categories to illustrate our mechanism discussed
in Section 2.

3.1

Direct gauge mediation

Let us consider direct gauge mediation models [8, 9, 10, 11, 12] in which a subgroup of the
ﬂavor symmetry of the Intriligator-Seiberg-Shih (ISS) model [13] is gauged by SU (5)GUT .
The model is based on the SU (NC ) gauge theory with NF pairs of quarks QIα and antiquarks Q̃αI . Here, I and α run from I = 1 to I = NF and from α = 1 to α = NC ,respectively. We assume, for simplicity, that they have a common mass
W = mQIα Q̃αI .

(6)We have a global ﬂavor symmetry SU (NF )F .
If the numbers of color and ﬂavor satisfy the relation NC + 1 ≤ NF < 23 NC , this theory
has a weakly coupled dual magnetic description at low energies. The dual magnetic theory
is described in terms of mesons ΦIJ and dual quarks ϕaI , ϕ̃Ia . Here, a = 1, · · · , ÑC is theindex of a dual gauge group SU (ÑC = NF − NC )mag . The superpotential of this theory5

is given by
W = hϕaI ΦIJ ϕ̃Ja − hµ2 ΦII .

(7)

Without a loss of generality, the Yukawa coupling constant h and the dimension one
parameter µ can be taken to be real and positive.
At the tree level, the equation of motion of Φ gives the F -term of Φ,−(FΦ† )JI = hϕaI ϕ̃Ja − hµ2 δIJ .

(8)

The right hand side of this equation cannot be zero, since the rank of the matrix ϕaI ϕ̃Ja
is no greater than NF − NC and the unit matrix δIJ has rank NF (> NF − NC ). Thus
some components of (FΦ† )JI are non-zero and SUSY is broken. If non-perturbative eﬀects
of SU (ÑC )mag are taken into account, however, SUSY is dynamically restored [13]. So
the SUSY breaking vacua are metastable.
Around the SUSY-breaking local minima of the potential, the ﬁelds ϕ, ϕ̃ and Φ canbe expanded like
ϕaI =

³

µδpa + δχap

δρai

´

Ã

,

ϕ̃Ia =µδap + δ χ̃pa
δ ρ̃ia

!

Ã

,

ΦIJ =

δYqp δ Z̃jp
δZqi δΦij

!

,

(9)

where p = I for 1 ≤ I ≤ NF − NC and i = I for NF − NC + 1 ≤ I ≤ NF . These vevs
break the global ﬂavor symmetry SU (NF )F down to SU (NF − NC )F × SU (NC )F . To
make this model a direct gauge mediation model, we embed the SU (5)GUT gauge group
into a subgroup of SU (NC )F or SU (NF − NC )F . Let us consider the theory above themass scale µ for each case [8].
1. If SU (5)GUT ⊂ SU (NC )F , in the magnetic theory, ﬁelds charged under SU (5)GUT
are (a part of) δρ, δ ρ̃, δZ, δ Z̃ in the (anti-)fundamental representation of SU (NC )Fand δΦ in the adjoint representation of SU (NC )F . Then, the contribution to the β(mag)= 2(NF − NC ) + NC =function of the SU (5)GUT gauge coupling is given by Nmess

2NF − NC 1 . (Note that the adjoint representation of SU (NC )F decomposes into
an adjoint representation of SU (5)GUT , NC − 5 ﬂavors of fundamental and antifundamental representations of SU (5)GUT , and some singlets.) In the electric theory
1

Nmess is not equal to the one contributing to the gaugino and sfermion soft masses. In this paper weare deﬁning Nmess only by the contribution to the gauge coupling β function.

6

(ele)
Nmess= NC . From the inequalities NC ≥ 5 and NC + 1 ≤ NF < 23 NC , we obtain
(mag)
(ele)
Nmess
≥ 7 and Nmess≥ 5.

2. If SU (5)GUT ⊂ SU (NF −NC )F , in the magnetic theory, ﬁelds charged under SU (5)GUT
are (a part of) δχ, δ χ̃, δZ, δ Z̃ in the (anti-)fundamental representation of SU (NF −NC )F and δY in the adjoint representation of SU (NF − NC )F . (This counting is applicable above the mass scale µ. Below µ, the SM gauge group is in the diagonal
subgroup of SU (NF −NC )F ×SU (ÑC )mag .) Then, the contribution to the β function(mag)of the SU (5)GUT gauge coupling is given by Nmess= (NF −NC )+NC +(NF −NC ) =(ele)
2NF − NC . In the electric theory Nmess= NC . From the inequalities NF − NC ≥ 5
(mag)
(ele)
> 20 and Nmess> 10.
and NC + 1 ≤ NF < 23 NC , we obtain Nmess

In the case SU (5)GUT ⊂ SU (NF − NC )F , Nmess is too large, so we concentrate on thecase SU (5)GUT ⊂ SU (NC )F . In this model, SUSY breaking is mediated to the MSSM
sector by the ﬁelds δρ, δ ρ̃, δZ, δ Z̃. The superpotential becomes
W = hϕaI ΦIJ ϕ̃Ja − hµ2 ΦII= hµ(δρai δZai + δ ρ̃ia δ Z̃ia ) + hρai Φij ρ̃ja − hµ2 Φii + · · · ,(10)where dots represent terms irrelevant for the gauge mediation. Φii has a non-vanishing
F -term, and then δρ and δ ρ̃ have SUSY-breaking masses. This is the type of gauge
mediation studied in Ref. [14]. R-symmetry breaking is rather non-trivial in this modeland one has to consider some modiﬁcation of the theory. See [8, 9, 10, 11, 12] for details.
Even in the case SU (5)GUT ⊂ SU (NC )F , Nmess ≥ 5 in both the electric and magnetic
theory. In fact, the messenger number Nmess is smaller in the electric theory than in
the magnetic theory, which was considered as a solution to the Landau pole problem in
Ref. [15]. However, the analyses of Ref. [2] suggest that the two-loop eﬀects from the
MSSM sector make it diﬃcult to maintain the perturbative GUT uniﬁcation. Furthermore, the messenger ﬁelds are charged under the strong hidden gauge group SU (NC ) in
the electric theory 2 . Thus as explained in Section 2, this model suﬀers from the severe2

In the magnetic theory, SU (5)GUT charged ﬁelds have both the SU (ÑC )mag gauge interaction (ifÑC ≥ 2) and the Yukawa interaction in the superpotential (7). Then it is non-trivial whether the total
eﬀect of these interactions decreases or increases the eﬀective messenger number, compared with the naivemagmessenger number Nmess.

7

Landau pole problem when the messenger mass scale is of order 105 GeV. Notice thatsuch a small mass ∼ 105 GeV for the messenger is required in the models [9, 10, 11],
since MSSM gaugino masses vanish at the leading order of SUSY-breaking scale (seealso Ref. [16]). Thus the SUSY-breaking scale and the messenger mass scale must becomparable, of order 105 GeV.
We now consider a modiﬁcation of the model to avoid the Landau pole problem. In
the electric theory, we add a chiral ﬁeld Aαβ which transforms in the adjoint representationof SU (NC ) gauge group. We introduce the new terms in the superpotential
W ⊃√2λQiα Aαβ Q̃βi + mA Aαβ Aβα ,(11)where i = NF − NC + 1, · · · , NF . This is in fact the model considered in Section 2,
with the identiﬁcation Pαp |Section 2 = Qpα (p = 1, · · · , NF − NC ), NQ |Section 2 = NC and
NP |Section 2 = NF −NC . We take NC = 5 and 2 ≤ NF −NC ≤ 4 in the following discussion.
The dynamics of the model is as follows. At high energies, we assume that the theory
is near the conformal ﬁxed point. Then, as discussed in Section 2, the eﬀective messengereﬀ
number Nmessis smaller than 5 (see Table 1). Below the mass scale mA , the adjoint ﬁeld

A decouples from the dynamics, and the theory exits from the conformal ﬁxed point andthe conﬁnement occurs. Then at the low energies the model can be described by the
weakly coupled magnetic theory. SUSY is broken as in the ISS model, and the direct
gauge mediation works.
However, the low energy theory is not completely the same as the original ISS model.
Integration of the adjoint ﬁeld A generates a superpotential
W ⊃−

·

¸

·

¸

λ2
1
λ2 Λ2
1
(Qiα Q̃αj )(Qjβ Q̃βi ) −(Qiα Q̃αi )(Qjβ Q̃βj ) = −δΦij δΦji −δΦi δΦj ,
2mA
NC
2mA
NC i j(12)

where Λ is the conﬁnement scale of the electric theory deﬁned by Qiα Q̃αj = ΛΦij . When
NC = NQ |Section 2 = 5, this term gives mass to the traceless part of δΦ, i.e. the part whichtransforms in the adjoint representation of SU (NC )F = SU (5)GUT . The traceless part ofδΦ does not take part in SUSY breaking and gauge mediation, but this ﬁeld contributes
to the β functions of the SM gauge coupling constants. So below the mass scale λ2 Λ2 /mA ,the “messenger number” contributing to the β function is Nmess = 2(NF − NC ). Thus,8if we take NF − NC = 2, the messenger number Nmess is smaller than 5 for the energy
scale below λ2 Λ2 /mA in the magnetic theory. On the other hand, above the scale mA , theeﬀtheory is electric and Nmessis small because of the mechanism of Section 2. Furthermore,Λ is roughly related to mA by the equation Λ ∼ mA exp(−8π 2 /(3NC − NF )g∗2 ), where g∗is the gauge coupling constant of SU (NC ) at the ﬁxed point. Then, if the ﬁxed pointis strongly coupled, which is the case in the present model, mA and λ2 Λ2 /mA are of thesame order. Thus the dangerous energy scale between λ2 Λ2 /mA and mA is narrow andthe perturbative uniﬁcation of the SM gauge couplings is maintained.
In the case 2 < NF − NC ≤ 4, Nmess is larger than 5 in the magnetic theory. However,
if we take mA suﬃciently low to ensure that the theory is in the electric theory over awide range of energy scale, the Landau pole may be avoided. For example, consider thecase NF − NC = 3. For simplicity, we use the following approximation; we approximate
eﬀ(ele)λ2 Λ2 /mA ∼ mA , and above the scale mA , Nmess= 4.2 taken from Table 1 and beloweﬀ(mag)the scale mA , Nmess= 2(NF − NC ) = 6. Furthermore, we suppose that all the MSSM

sparticles have masses not far from the Z boson mass, mZ ' 91 GeV. Then, the SM
QCD coupling constant g3 at the GUT scale MGUT ∼ 1016 GeV is , at the one-loop level,
given by
µ

¶

µ

Mmess
mA
8π 2
8π 2
eﬀ(mag)'+ 3 log+ (3 − Nmess
) log
2
2
g3 (MGUT )g3 (mZ )
mZ
Mmess
µ
¶
MGUT
eﬀ(ele)+(3 − Nmess
) log
,
mA

¶

(13)

where Mmess is the messenger mass scale which we take Mmess ∼ 105 GeV for a low-scalegauge mediation, and 8π 2 /g32 (mZ ) ' 53.2 from experiments. Requiring g32 (MGUT )/4π <∼ 1,
we obtain the constraint on mA as
mA < 108 GeV.
∼

(14)

Thus, mA can be taken much larger than Mmess ∼ 105 GeV.

3.2

Semi-direct gauge mediation

We consider a SUSY-breaking model based on an SU (5)hid gauge symmetry. It is known [17]that the SUSY is broken when we introduce only two matter multiplets, V α and Xαβ in

9the representations 5∗ and 10 of SU (5)hid , respectively. We now introduce NF pairsof fundamental quarks Qiα and anti-quarks Q̃αi . Here, i = 1, · · · , NF and α = 1, · · · , 5
and they belong to (N∗F , 5) and (NF , 5∗ ) representations of the SU (NF )F × SU (5)hid ,respectively. We introduce a common bare mass term for the messengers, for simplicity,
W = mQ Qiα Q̃αi .

(15)

We gauge a subgroup of SU (NF ) under the GUT gauge group [18]. This is the set
up for the semi-direct gauge mediation in the SU (5)hid SUSY-breaking model 3 . The
bi-fundamental messenger ﬁelds Q and Q̃ link the SU (5)hid hidden gauge sector and the
MSSM sector, thus SUSY-breaking is mediated to the MSSM sector. In particular, when
NF ≥ 6, it can be shown that the theory has a conformal ﬁxed point above the mass scale
mQ . Then, after the decoupling of the messengers Q and Q̃, the theory exits from theconformal ﬁxed point and the SUSY breaking occurs. This is a conformal gauge mediation
model proposed in Ref. [18]. However, we only impose NF ≥ 5 in this paper.
In the above model, the messenger ﬁelds are charged under the strong SU (5)hid gaugeeﬀgroup. Thus, as discussed in Section 2, the eﬀective messenger number Nmessis larger

than 5. Thus this theory suﬀers from the Landau pole problem. To avoid the problem,
we introduce a chiral ﬁeld Aαβ transforming in the adjoint representation of SU (5)hid . We
introduce a superpotential,
W ⊃√2λQiα Aαβ Q̃βi + mA Aαβ Aβα ,(16)for the mechanism explained in Section 2 to work. In fact, this model is not the sameas the example model described in Section 2, but we can study the conformal ﬁxed pointof this theory by using a-maximization technique explained in Appendix B. The result islisted in Table 2. One can see that there is no Landau pole problem for NF = 5, 6 and 7.
3

In semi-direct gauge mediation, a messenger number is not necessarily larger than or equal to 5.
For example, in the models of Ref. [5], the messenger number is minimally 2, so there is no Landaupole problem. However, in semi-direct gauge mediation, sparticle masses (especially gaugino masses) aresuppressed by hidden sector loops [18, 5], so if one wants to bulid a model in which the gravitino is very
light (< O(10) eV), the hidden sector gauge theory should be strongly coupled. Because the gauge theory
should be strongly coupled even when we add messenger ﬁelds, the hidden sector gauge group should
be somehow large, as in the above SU (5)hid model. Then the model may suﬀer from the Landau pole
problem.

10

NF
5
6
7

eﬀ
γQ
γA
γV
γX
Nmess
0.264 −0.528 −0.656 −0.901 3.68
0.160 −0.320 −0.527 −0.731 4.20
0.063 −0.126 −0.308 −0.438 4.69

Table 2: The anomalous dimensions of the ﬁelds of the model. γQ is the anomalous
dimension of Q and Q̃. γA , γV and γX are the anomalous dimensions of A, V and X
respectively. The fact that the anomalous dimensions of Q and Q̃ are the same is noteﬀ
obvious because the model is chiral, but a-maximization shows that is the case. Nmessiseﬀ
deﬁned by Nmess = 5(1 − γQ ).

The dynamics of the model is as follows. We take mA > mQ . Then above the mass
mA , we assume that the theory is near the conformal ﬁxed point. Below the threshold ofA, the SU (5)hid gauge coupling becomes larger, and it blows up (when NF = 5) or goesto another ﬁxed point discussed above (when NF ≥ 6). In any case, after the decouplingof the messenger ﬁelds Q and Q̃, SUSY is broken [19, 17].
In fact, the low energy theory after the decoupling of A is not the same as the originalsemi-direct gauge mediation of Ref. [18]. As in Eq. (12) of the previous subsection, theintegration of A generates a superpotential
¸

·

λ2
1
W ⊃−
(Qiα Q̃αj )(Qjβ Q̃βi ) − (Qiα Q̃αi )(Qjβ Q̃βj ) .
2mA
5

(17)

The presence of this superpotential is very interesting, since this term explicitly breaks
R symmetry of the original semi-direct gauge mediation model, which may be useful fora generation of gaugino masses [20]. On the other hand, however, this term generates
D

E

SUSY preserving vacua at QQ̃ ∼ mA mQ /λ2 , making the SUSY breaking vacuum at
DE

QQ̃ = 0 metastable.

3.3

Composite messenger model

We consider a strong SU (5)hid gauge theory with 5 pairs of fundamental quarks Qiα andanti-quarks Q̃αi . Here, i and α run from 1 to 5 and they belong to (5∗ , 5) and (5, 5∗ )representations of the SU (5)GUT × SU (5)hid , respectively. Those 5 pairs of quarks play arole of messengers. We take a superpotential for the messengers,
W = hXQiα Q̃αi ,
11

(18)

where X = M +F θ2 is a SUSY-breaking spurion ﬁeld. This model is the so-called minimal
gauge mediation (see [3] and references therein), aside from the fact that the messengers
are charged under the strong gauge group SU (5)hid .
The reason that we introduce the SU (5)hid gauge interaction is to conﬁne the messenger quarks and anti-quarks, Q and Q̃, forming composite ﬁelds. One of the composite
states can be a candidate for the dark matter of the universe [7]. In fact we have, at low
energies, mesons
Mji = Qiα Q̃αj ,(19)

and baryons
B = det Q,

B̃ = det Q̃,(20)with the constraint
det M − B B̃ = Λ10 ,(21)

where Λ is the dynamical scale of SU (5)hid . The baryon B and anti-baryon B̃ are longlived, since we have an approximate baryon number conservation [7]. Then, they can bea candidate for the dark matter.
Above the energy scale Λ, the messenger number is 5, and because the messengers areeﬀ
charged under the strong gauge group, the eﬀective messenger number Nmessis larger than

5 as explained in Section 2. Below the energy scale Λ, the traceless part of Mji transforms
in the adjoint representation of SU (5)GUT , so the messenger number (in the deﬁnition ofthis paper) is also 5. Thus this model suﬀers from the Landau pole problem.
For the mechanism of Section 2 to work, we introduce additional NP ﬂavors of quarks
Pαp and P̃pα in the representation 5 and 5∗ of SU (5)hid , respectively. Here p is the ﬂavor
index, p = 1, · · · , NP . We further introduce an adjoint ﬁeld Aαβ of SU (5)hid , and introcuce
a superpotential
W ⊃√2λQiα Aαβ Q̃βi + mP Pαp P̃pα + mA Aαβ Aβα ,(22)for the additional ﬁelds, as in Section 2. Then the messenger model becomes the sameas the model in Section 2. The eﬀective messenger number above the mass mP and mA

12

is given in Table 1 with NC equal to 5. We see that the Landau pole problem can be
avoided by taking mP and mA appropriately small.
The dynamics of the model is as follows. We assume that mP and mA are of thesame order, mP ∼ mA , for simplicity, and the theory is near the conformal ﬁxed point
above the threshold of these ﬁelds. After the decoupling of P, P̃ and A, the SU (5)hidgauge coupling becomes strong and the gauge theory conﬁnes the color degrees of freedom,making composite ﬁelds described above.
However, we have to take care of the following point. When the theory is on theconformal ﬁxed point, the Yukawa coupling in Eq. (18) becomes smaller as we lower the
renormalization scale. Suppose that the theory is on the conformal ﬁxed point from theenergy scale M∗ down to m∗ (∼ mA ∼ mP ). Then, neglecting all eﬀects other than theﬁxed point dynamics, the Yukawa coupling h at the scale m∗ ish|m∗ ∼

µ

m∗
M∗

¶ γQ

h|M∗ .

(23)

We show that the requirement m3/2 < 16 eV [1] leads to a constraint on NP . For
the messenger quarks not to be tachyonic 4 , the SUSY-breaking scale F must satisfy
hF < (hM )2 . Then, the gaugino mass is constrained as
Mg̃ ' n

α hF
α√
hF ,
<n
4π hM
4π

(24)

where α is the SM gauge coupling ﬁne structure constant corresponding to the gaugino g̃,and n is a “messenger number” contributing to the gaugino masses (in the present model,
n = 5). Then, the gravitino mass is constrained asÃF(4πα−1 n−1 Mg̃ )3.4 × 10−3√
m3/2 = √>= 16 eVh3MP l3hMP l2!Ã

α−1 Mg̃60 100 GeV

!2

,

(25)

where MP l ' 2.4 × 1018 GeV is the reduced Planck mass. Note the dependence 1/h ofthis lower bound. Thus, to achieve the light gravitino mass, h should not be too small,and thus γQ should not be too large. In particular, the model with NP = 2 may not be
eﬀ
is the smallest in this case.
favored, although the eﬀective messenger number Nmess
4

The messenger ﬁelds are strongly coupled in the present model, but here we pretend as if they canbe treated as weakly coupled fundamental quarks, for simplicity. In fact, it is known that the gaugino
masses are not so aﬀected by strong interactions, due to the gaugino screening mechanism [21].

13Acknowledgement
We would like to thank S. Shirai for a suggestion crucial for this work, and for very useful
discussions and comments. This work was supported by World Premier International
Center Initiative (WPI Program), MEXT, Japan. The work of KY is supported in partby JSPS Research Fellowships for Young Scientists.

Appendix A

Eﬀective messenger number in asymptotically free gauge theories

In this appendix we study how strong gauge interactions make the Landau pole problemof the SM gauge coupling severe. Suppose that messenger ﬁelds transform under the
representation r + r̄ of some strong gauge group G. At the one-loop level, the gauge
coupling g of the gauge group G is given by
8π 2
8π 2
= 2 + b log(µ/M0 ),g 2 (µ)g0(A.1)

where b is the coeﬃcient of the one-loop β function, and g0 is the gauge coupling at thescale M0 . We deﬁne t ≡ log(µ/M0 ) and g 2 (µ)/8π 2 ≡ hg (µ) for simplicity. Then the above
equation is rewritten as−1
hg (t) = (h−1g0 + bt) .

(A.2)

The anomalous dimension of the messenger ﬁelds is, at the one-loop level, given by−1γ(t) = −2C2 (r)hg (t) = −2C2 (r)(h−1
g0 + bt) ,(A.3)

where C2 (r) is the quardratic Casimir of the representation r.
Taking into account Eq. (1), we deﬁne the averaged value of the anomalous dimension
between µ = M0 and µ = M1 as
1 Z t1
2C2 (r) log(1 + bhg0 t1 )γ̃ ≡dtγ(t) = −
,
t1 0
b
t1

(A.4)

where t1 = log(M1 /M0 ). Then the averaged eﬀective messenger number is
eﬀ
Ñmess= (1 − γ̃)Nmess ,

14

(A.5)

where Nmess is “the tree level value” of the messenger number.
For example, consider the case G = SU (5), r = 5, b = 3 · 5 − 5 = 10, M0 = Mmess ∼10 GeV, and M1 = MGUT ∼ 1016 GeV. If we further assume that the coupling is very
6

strong at M0 , e.g. γ(µ = M0 ) ' −1, we obtaineﬀ= (1 − γ̃)5 ' 5.40.
Ñmessγ̃ ' −0.080,

(A.6)

eﬀ
The SM gauge couplings receive a contribution Ñmesslog(MGUT /Mmess ) from the messen0
ger ﬁelds. Deﬁning Meﬀ
by the equationeﬀ
0Ñmesslog(MGUT /Mmess ) = Nmess log(MGUT /Mmess
),

(A.7)

we obtain
µ

eﬀ

Mmess
MGUT
=
0
Mmess
Mmess

mess −1
¶ Ñ
N
mess

∼ 6.

(A.8)

The lower bound on the messenger mass scale becomes larger by this factor due to strong
gauge interactions.

Appendix B

Details on conformal ﬁxed point

In this appendix, we describe how to ﬁnd a conformal ﬁxed point and compute anomalous
dimensions of matter ﬁelds, taking the model of Section 2 as an example. We do not
restrict the number of ﬂavors of Q, NQ , equal to 5 in this appendix.
Let us ﬁrst discuss the existence of the infrared ﬁxed point in the model in Section 2.
For a time being we consider only the theory where the perturbative calculation forthe β function is reliable. At the perturbative level, one can discuss the existence of aconformal ﬁxed point by explicitly considering the renormalization group equations, as
was ﬁrst done in Ref. [22]. The NSVZ β function of the gauge coupling g and the βfunction of the Yukawa coupling λ of the model in Section 2 are given byg 4 3NC − (1 − γA )NC − (1 − γQ )NQ − (1 − γP )NP∂ 2
g =− 2,∂µ8π
1 − NC g 2 /8π 2∂= µ λ2 = (γA + 2γQ )λ2 ,∂µβg = µ

(B.1)

βλ

(B.2)

15

where we have taken λ to be real without a loss of generality, and γA , γQ and γP are theanomalous dimensions of A, Q, and P respectively. At the one-loop level, they are given
by 5
γA '

NQ λ2 − 2NC g 2
,
8π 2

γQ '

NC2 − 1 λ2 − g 2
,
NC
8π 2

γP ' −

NC2 − 1 g 2.
NC 8π 2

(B.3)

By taking a large N limit, NC , NQ , NP À 1 with NQ /NC ∼ O(1), NP /NC ∼ O(1) andn ≡ 3NC −NC −NQ −NP ∼ O(1), one can ﬁnd a solution to the equations βg = 0, βλ = 0
with the couplings g and λ being very small (Banks-Zaks like ﬁxed point). The result is
4n
λ2
'
,
2
2
8π
(2NC − NQ ) + NP (2NC + NQ )NQ + 2NC λ2
g2
'
,
8π 2
4NC
8π 2

(B.4)

and
γQ '

(2NC − NQ )n(2NC + NQ )n, γP ' −, (B.5)
2(2NC − NQ ) + NP (2NC + NQ )(2NC − NQ )2 + NP (2NC + NQ )with γA = −2γQ . It is also easy to check that the ﬁxed point is infrared stable. The
coupling constants (or more precisely, ’t Hooft couplings) of the theory are small, so we
can trust perturbative calculation. Thus we consider that the existence of a conformal
ﬁxed point in the above limit is established, and the ﬁxed point values of the anomalous
dimensions are given by Eq. (B.5).
However, we are interested in the case where the coupling is strong, so that the anomalous dimension γQ is quite large and any perturbative calculation is not reliable at all. A
very astonishing fact of supersymmetric conformal ﬁeld theory is that anomalous dimensions of fundamental ﬁelds can be determined exactly, even in strongly coupled theories.
The general method is called a-maximization [24]. In N = 1 superconformal ﬁeld theories,
there is an R symmetry which appears in superconformal algebra (which is an extensionof ordinary supersymmetry algebra). In some theories there may be a unique anomaly
free R symmetry, and if the theories are in the conformal window [25], the R symmetry
must be the one which appears in the superconformal algebra. However, in general there
is a family of anomaly free R symmetries (as in the model of Section 2; see below), and5

See e.g. Section 5.5 of Ref. [23]. Note that our convention for the anomalous dimension is larger thanthat used in Ref. [23] by factor 2.

16

R

A
Q, Q̃
P, P̃
−2x 1 + x 1 + NP−1 (2NC − NQ )x

Table 3: R charges of the ﬁelds. x is a parameter that parametrize the ambiguity of thedeﬁnition of R symmetry.

we cannot determine from symmetry argument alone which R symmetry is the superconformal one 6 . According to Ref. [24], the superconformal R symmetry is the one which(locally) maximizes the following combination of t’ Hooft anomalies,X[3(Ri − 1)3 − (Ri − 1)],

(B.6)

i

where the sum is taken over fermions of a theory and Ri − 1 is the R charge of thefermion in chiral ﬁeld i. This condition determines the R charge Ri of the chiral ﬁeld i.
Furthermore, the scaling dimension Di and the anomalous dimension γi of chiral ﬁeld i isrelated to the R charge Ri by the equation
1+

γi
3= Di = R i .
2
2

(B.7)

The ﬁrst equality in Eq. (B.7) is almost the deﬁnition of the anomalous dimension in
conformal ﬁeld theory. For the second equality, see e.g. [26]. From Eq. (B.7), we candetermine the anomalous dimension γi from the R charge Ri .
Let us apply the above method to the model of Section 2. We, here, neglect all masses
for the ﬁelds, and assume that the model is in the conformal window for certain valuesof NC , NQ and NP . The R charges of the ﬁelds are shown in Table 3. We have imposedthat Q and Q̃ (P and P̃ ) have the same R charge. Even then, the R charges of the ﬁelds
are not uniquely determined. We have parametrized the ambiguity of R charges by x.
Then, we deﬁne the following function of x,a(x) ≡

X[3(Ri − 1)3 − (Ri − 1)]i= (NC2 − 1)[3(−2x − 1)3 − (−2x − 1)] + 2NC NQ [3x3 − x]+2NC NP [3(NP−1 (2NC − NQ )x)3 − (NP−1 (2NC − NQ )x)].
6

(B.8)

In this paper we do not consider the case where the superconformal R symmetry is an accidental
symmetry of a low energy theory.

17

NC = 5NC = 6NC = 7NC = 8NC = 9NP = 2
0.273
0.422
×
×
×

NP = 3 NP = 4 NP = 5 NP = 6
0.143
0.059
×
×
0.280
0.179
0.104
0.046
0.391
0.287
0.205
0.138
0.478
0.376
0.292
0.223
×
0.448
0.366
0.296

Table 4: The values of γQ obtained in Eq. (B.5) by perturbative calculation. This tableshould be compared with Table 1, where the exact value of γQ is listed. NQ is taken to
be 5.

Then, the condition for the local maximization of a(x) is given by
∂a(x)
= 0,
∂x

∂ 2 a(x)
< 0.
∂x2

(B.9)

Solving these equations is quite straightforward. Using the solution for x, the anomalous
dimesion of e.g. Q is given by γQ = 3RQ − 2 = 3x + 1. The value of γQ is listed in Table 1.
As a check, we list the one-loop value of γQ obtained in Eq. (B.5) in Table 4. Note thatthe agreement between Table 1 and 4 is quite good, and becomes better as the coupling
becomes weaker.
For what values of (NC , NQ , NP ) the model has a conformal ﬁxed point is a rather nontrivial question. In conformal ﬁeld theory, it is known that all gauge invariant operatorsof a theory have scaling dimensions greater than or equal to 1 [27]. As a criterion of theexistence of a conformal ﬁxed point, we require that all gauge invariant chiral (primary)operators have scaling dimensions greater than or equal to 1. Such a criterion was ﬁrst
used in Ref. [25] to ﬁnd a conformal ﬁxed point in SUSY QCD. Especially, in the caseof the present models, we have imposed that the scaling dimensions of gauge invariant
chiral operators tr A2 and P a P̃b satisfy the conditions
µ

¶

γA
≥ 1,
2
µ
¶
γP
= 2 1+≥ 1.
2

Dtr A2 = 2 1 +(B.10)DP P̃

(B.11)In this paper we assume that if these conditions are satisﬁed, the model has a conformal
ﬁxed point.

18

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20