YITP-10-10

Gaugino Mass and Landscape of Vacua

Yuichiro Nakai1 and Yutaka Ookouchi2,3
1

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
2

Perimeter Institute for Theoretical Physics, Ontario N2L2Y5, Canada
3

Institute for the Physics and Mathematics of the Universe (IPMU),
University of Tokyo, Chiba, 277-8583, Japan

Abstract
In direct gauge mediation, gaugino masses often vanish at the leading order of supersymmetry breaking. Recently, this phenomenon is understood in connection with the global
structure of vacua in O’Raifeartigh type models. In this note, we further explore a connection
between gaugino masses and the landscape of vacua in a variety of gauge mediation models.
In particular, we present a calculable model with non-vanishing leading order gaugino masseson the lowest energy vacuum.

1

Introduction

Gauge mediation [1] is a highly predictive and an attractive way of transmitting supersymmetry breaking of a hidden sector to the supersymmetric standard model. Many gauge mediation
models are already known (See [2, 3] for reviews). Among them, direct gauge mediation is
the model that the ﬂavor symmetries in a hidden SUSY breaking sector are weakly gauged
and identiﬁed as the standard model gauge symmetries. In most controllable direct gauge√
mediation models, the scale of supersymmetry breaking F (the square-root of the F-termof a SUSY breaking ﬁeld) is much smaller than the messenger scale M . In such models, theleading contributions to the standard model gaugino masses in an F/M 2 expansion can becalculated by a powerful tool, analytic continuation into superspace [4, 5]. However, it often
happens in direct gauge mediation models that the leading contributions to gaugino masses
vanish even if R-symmetry is broken, ﬁrstly emphasized in [6].1 Since there are no such cancellations for scalar masses, this implies that the gauginos are much lighter than the scalars.2
Then, we cannot obtain order 1 TeV gaugino masses and scalar masses at the same time,
which causes the standard model hierarchy problem again. Furthermore, recent experimental
data severely contrain such a model with the light gauginos [10]. Thus, it is an interesting
challenge to generate sizable gaugino masses in direct gauge mediation. The authors of [11]avoided this issue by exploiting a higher energy metastable state (This possibility was recently emphasized in [12, 13]). However, the reason why an uplifted vacuum generates sizable
gaugino masses has remained mysterious.
Recently, Komargodski and Shih (KS) shed light on this curious feature of direct gauge
mediation and clariﬁed that the pseudomoduli space cannot be locally stable everywherein order to generate sizable gaugino masses [14]. It is interesting that anomalously small
gaugino masses are closely related to the global structure of vacua. It is worth exploring thisconnection further in more general situations.
In this paper, we reconsider the arguments of [14] in a variety of gauge mediation models
and provide a way to generate the leading order gaugino masses while keeping all messenger
directions stable everywhere. Furthermore, we construct a gauge mediation model with nonvanishing leading order gaugino masses on the lowest energy vacuum.
The rest of the paper is organized as follows. In section 2, we will review a property in
O’Raifeartaigh type models with canonical Kähler potential and a connection between the1

Anomalously small gaugino mass problem also occur in another class of gauge mediation models suchas semi-direct gauge mediation [7]. In this class, sizable gaugino masses can be generated by introducing astrongly coupled messenger sector [8].
2
We can suppress the scalar masses by exploiting a superconformal dynamics [9].

1global structure of vacua and the leading order gaugino masses. In section 3, we will show how
non-canonical Kähler potential of messenger ﬁelds alters the stability of messenger directions
at a point of the pseudomoduli space. We also provide an explicit model which has non-zero
leading order gaugino masses and a pseudomoduli space that is locally stable everywhere.
In section 4, we present a calculable model with an additional U (1) gauge interaction which
generates the leading order gaugino masses on the lowest energy vacuum.

2

Review of the KS theorem

First, we will brieﬂy describe the existence of a pseudomoduli space discussed in [14, 15] by
exploiting a simple example. Let us consider the following Wess-Zumino model with canonicalKähler potential presented in [16],W = λX(φ1 φ˜1 + φ2 φ˜2 ) + mφ1 φ˜2 + f X(2.1)

where X is a SUSY breaking ﬁeld and φ1 , φ˜1 , φ2 , φ˜2 are vector-like pairs of messenger ﬁelds
charged under the standard model gauge symmetries. We can choose parameters λ, m, f asreal without loss of generality. On the metastable SUSY breaking vacuum, hφ1 i = hφ˜1 i =hφ2 i = hφ˜2 i = 0 and hXi is the pseudomoduli space of vacua. That is, it takes an arbitraryvalue at the classical level, but takes a deﬁnite value by quantum eﬀects. In this model,
as discussed in [16], the vev hXi takes a non-zero value, so R-symmetry is spontaneously
broken. As this example, every O’Raifeartaigh type model with canonical Kähler potential
has a pseudomoduli space on every SUSY breaking vacuum.
Next, we will see a connection between the global structure of vacua and the leading ordergaugino masses by dealing with the same example as above. The leading order gaugino masses
are calculated by analytic continuation into superspace technique such as,mg̃ ∼ f∂log det MF ,
∂X

(2.2)

where MF is the fermion mass matrix of messengers. In this model, the determinant of thefermion mass matrix is given by det MF = λ2 X 2 and has an X dependence, so gaugino masses
are generated at the leading order of the SUSY breaking f ,
mg̃ ∼

f
.
hXi

(2.3)

On the other hand, in this model, there is a zero eigenvalue in(the fermion mass matrix
at)√hXi = 0. Here, the eigenvalues of the scalar mass matrix are m2 ± m4 + 4λ2 f 2 /2, so
there is a tachyonic direction. As this example, there is always at least one tachyonic direction
2

on the pseudomoduli space of vacua when gaugino masses are generated at the leading order.
Otherwise, the leading order gaugino masses vanish. For general arguments, see the original
paper [14].

3General cases

As we reviewed in the previous section, in every renormalizable O’Raifeartigh type model,
the pseudomoduli space cannot be stable everywhere to generate gaugino masses. However,a renormalizable model is not always a good description of dynamical SUSY breaking at
low-energy. In many SUSY breaking models, correction terms in Kähler potential are notnegligible. In this section, we emphasize that such terms aﬀect crucially the connection
between gaugino masses and the landscape of vacua.
3.1

Stability of messenger directions

Let us start with a general argument for the stability of messenger directions. Suppose wehave a superpotential interaction,
W = MF (X)ab φ̃a φb + f (X),

(3.1)

where X is a chiral superﬁeld which is responsible of SUSY breaking and φ, φ̃ are messengers.
The lower indecies of the messenger mass matrix MF denote the derivatives with respect tomessenger ﬁelds. When we turn on a generic non-canonical Kähler potential, X direction
is not necessarily pseudo-ﬂat as discussed in [15].3 Nevertheless, in order to focus on thestability of messenger directions at a point of the pseudomoduli space like [14], we can keepa ﬂat direction by imposing the following condition on the metric [17],∂X g X X̄ 0 = 0,

(3.2)

where |0 denotes hφa i = hφ̃a i = 0. It is easy to check that the scalar potential with thiscondition keeps X direction ﬂat.
In this setup, the boson mass-squared matrix of the messengers is given by()∗(M∗F MF )ab̄ − Aab̄
Fab
2
.
MB =
Fāb̄
(MF M∗F )āb − Aāb

(3.3)

Here,
Fab = FX∗ (∂X MF )ab ,3

Aab̄ = Rab̄ X X̄ |FX |2 ,Also, D-term SUSY breaking models do not always have pseudomoduli spaces.

3

(3.4)

where Rab̄ X X̄ are components of the Riemann tensor. We simply assumed gX X̄ = 1 at
φ̃a = φb = 0. Suppose va is a unit vector satisfying (MF )ab vb = 0. Then, a bosonic mode
corresponding to this direction has a mass,()( † T) 2v
v v MB= v T Fv − v † Av + c.c..
(3.5)
v∗
If Av = 0 or simply if A = 0, then the bosonic mode must be massless in order to have aconsistent vacuum, or we have to allow the vacuum to have a tachyonic direction. However,in general, this does not true. As we will demonstrate below, one can easily lift a tachyonicdirection and make the pseudomoduli space stable everywhere by using the contribution fromthe non-canonical part of Kähler potential A.
3.2

A model with non-canonical Kähler potential

In the rest of this section, we will try to construct the model which has non-zero leading
order gaugino masses and a pseudomoduli space that is locally stable everywhere. Let us
consider the model discussed in section 2 with non-canonical Kähler potential, that is, its
superpotential is given by
W = λX(φ1 φ˜1 + φ2 φ˜2 ) + mφ1 φ˜2 + f X.

(3.6)

Notation is explained in the previous section. This model with canonical Kähler potential
has a tachyonic direction around hXi = 0, so we will try to lift this direction by introducing
non-canonical Kähler potential,4
)
)
(
) (
)
|X|2 (|X|2 (
2
2
2
2
2
|φ
|
+
|
φ̃
|
+
1
−
|
φ̃
|
+
|φ
|
,
(3.7)
K = |X| + 1 +
1
2
1
2
M2
M2
where M is a large cut-oﬀ scale of the theory and we have required vanishing messenger mass
supertrace so that our model is UV insensitive [18, 19]. Since the above Kähler potential
satisﬁes the condition (3.2) given in the previous subsection, the pseudo ﬂat-direction of X
is kept. There is a zero eigenvalue in the fermion mass matrix at hXi = 0, and here theeigenvalues of the boson mass-squared matrix of messengers are)1( 2 √ 4m ± m + 4 λ2 f 2 − 4 (f /M )2 m2 + 4 (f /M )4 .
(3.8)
2
We can impose a condition between parameters of the model such as λ2 f 2 − (f /M )2 m2 +(f /M )4 < 0 so as not to have any tachyonic direction. Since non-canonical Kähler potential
4

When the sign of non-canonical part of Kähler potential is positive, it is possible to ﬁnd a UV completionof this theory with extra massive chiral superﬁelds. On the other hand, when the sign is negative, we needother ingredients such as a massive vector boson for a UV completion [19].

4

of messengers does not contribute to the gaugino mass at the leading order, known as gaugino
screening [5], the leading order gaugino mass is given in the same fashion as the case withcanonical Kähler potential,5
f
mg̃ ∼
.
(3.9)
hXi
Here, the expectation value of X can be estimated by stabilizing the one-loop eﬀective potential. The Coleman-Weinberg potential in this kind of models has been calculated in [17],which claims that X does not stabilize at the origin and so R-symmetry is spontaneously broken even in the case with non-canonical Kähler potential. Therefore, we can obtain non-zero
leading order gaugino masses in the model with a pseudomoduli space that is locally stableeverywhere.
While we have considered a model with a pseudo-ﬂat direction, as we have seen in theprevious subsection, the existence of pseudomoduli is not guaranteed in models with noncanonical Kähler potential. In the next section, we will see the relation between this kind ofmodels and the landscape of vacua.

4

Sizable gaugino mass on the global minimum

When we consider the case where there is no pseudomoduli space, it becomes unclear how we
can generalize the statement of the KS theorem. We are interested in a connection between
the leading order gaugino masses and metastability of the vacuum. Then, we will try to solve
the question whether we can obtain non-vanishing gaugino masses on the global minimum ornot. The answer is yes. In [20], the authors obtained non-vanishing leading order gaugino
masses on the global minimum (See also [21] for a related work on this avenue realizing theultra-light gravitino mass). However, they used a dynamical SUSY breaking model and theresulting model is incalculable. Then, for our current purpose, we do not need to focus on
dynamical SUSY breaking, so we can take our familiar O’Raifeartaigh model.
The explicit model of the SUSY breaking sector is a U (1) gauge theory6 whose superpotential is given by
W = X0 (f + λϕ1 ϕ2 ) + m(X1 ϕ1 + X2 ϕ2 ).

(4.1)

The U (1) charge assignments of X0 , X1 , X2 , ϕ1 and ϕ2 are 0, −1, 1, 1 and −1 respectively.
We call this U (1) gauge interaction as the messenger gauge interaction. We can take all5

Unlike the gaugino mass, non-canonical Kähler potential of messengers contributes to the scalar mass at
the leading order.
6
The U (1) gauge coupling becomes strong at high energy. As a logical possibility, there may be a state
which has lower energy than the state we now consider by non-perturbative eﬀects. Then, for more rigorous
statement, we try to construct a gauge mediation model with the global minimum at least perturbatively.

5couplings, λ, m, f as real without loss of generality and assume f  m2 . On the SUSY
breaking vacuum, hX1 i = hX2 i = hϕ1 i = hϕ2 i = 0 and X0 has a non-zero F-term.
Next, consider the messenger sector. The simplest possibility for our purpose would be
the following,κ(4.2)
Wmess = yf Sf f˜ + yE SE Ẽ + S 3 ,3where f and f˜ are messengers charged under the standard model gauge symmetries andS, E, Ẽ are the standard model gauge singlets. Only E, Ẽ have charges 1, −1 under themessenger U (1) gauge interaction. We also take couplings yf , yE , κ as real. When we integrateout the SUSY breaking sector, two-loop correction generates positive scalar masses for ﬁelds
E and Ẽ like usual gauge mediation, which is given by( 2 )2 ( )2
gmess
λf
2
2
,
(4.3)
mE = mẼ ∼16π 2
mwhere gmess is the coupling of the messenger gauge interaction. As pointed out in [20], these
positive scalar masses generate negative mass squared by one-loop eﬀects of E and Ẽ such as−m2S '

4 2 2
Λ
yE mE ln
,
2
16π
mE

(4.4)

where Λ is the cut-oﬀ scale and we assume yE . 1 so that m2E  |m2S | is satisﬁed. Then, theeﬀective scalar potential of the messenger sector including these corrections is given by
Vmess =yE S Ẽ2+ yE SE22
2
2+ yf S f˜ + yf Sf + yE E Ẽ + yf f f˜ + κS 2

+m2E |E|2 + m2E |Ẽ|2 + m2S |S|2 .

(4.5)

This potential is minimized at
h|S|2 i =|m2S |, hf i = hf˜i = hEi = hẼi = 0.
2κ2

(4.6)

The contribution to the vacuum energy is given by
V0 = −

m4S
.
4κ2

(4.7)

This vacuum is the global minimum in certain parameter range. The standard model gaugino
mass can be calculated asmg̃ ∼h|FS |i
|mS |
= √ .
hSi
2

(4.8)

Therefore, we can obtain the leading order gaugino mass on the global minimum of thepotential, unlike direct gauge mediation without additional gauge interactions.
6While constructing a model with no phenomenological problem is not the purpose of thispaper, we ﬁnally comment on some points. Although the messenger U (1) gauge boson ismassless, it may cause no problem, since the standard model quarks and leptons do not havethe messenger U (1) charges. However, if we want to avoid the existence of additional masslessgauge bosons, we can higgs the gauge symmetry by adding the following superpotential,
W = hT (ΨΨ̃ − v 2 ),(4.9)

where Ψ, Ψ̃ are a vector-like pair of chiral superﬁelds charged under the messenger U (1) gaugeinteraction and T is a Lagrange multiplier ﬁeld. h is a coupling constant. This term does notrestore SUSY, so our argument in this section remains true even if we add this term to theabove model. Furthermore, in the SUSY breaking sector, there is a parity given by
X1 ↔ X2 , ϕ1 ↔ ϕ2 , E ↔ Ẽ, V ↔ −V,(4.10)and so the problematic FI term of the messenger U (1) gauge ﬁeld is forbidden.

Acknowledgments
We would like to thank M. Ibe, Z. Komargodski, D. Shih and T. Yanagida for useful discussions
and comments. YO’s research is supported by World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan. YN is grateful to the Perimeter Institute for
Theoretical Physics for their hospitality during the course of this work. YN and YO would
like to thank the Institute for advanced study for their hospitality. YO’s research at the
Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada
through NSERC and by the Province of Ontario through MRI.

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