Flavor violation in supersymmetric theories with gauged flavor symmetries
ABSTRACT In this paper we study flavor violation in supersymmetric models with gauged flavor symmetries. There are several sources of flavor violation in these theories. The dominant flavor violation is the tree-level $D$-term contribution to scalar masses generated by flavor symmetry breaking. We present a new approach for suppressing this phenomenologically dangerous effects by separating the flavor-breaking sector from supersymmetry-breaking one. The separation can be achieved in geometrical setups or in a dynamical way. We also point out that radiative corrections from the gauginos of gauged flavor symmetries give sizable generation-dependent masses of scalars. The gaugino mass effects are generic and not suppressed even if the dominant $D$-term contribution is suppressed. We also analyze the constraints on the flavor symmetry sector from these flavor-violating corrections. Comment: 22 pages, latex, 4 figures
arXiv:hep-ph/0211347v1 22 Nov 2002
Flavor violation in supersymmetric theories
with gauged flavor symmetries
Tatsuo Kobayashi,∗Hiroaki Nakano,†Haruhiko Terao,‡and Koichi Yoshioka§
∗Department of Physics, Kyoto University, Kyoto 606-8502, Japan
†Department of Physics, Niigata University, Niigata 950-2181, Japan
‡Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan
§Department of Physics, Tohoku University, Sendai 980-8578, Japan
In this paper we study flavor violation in supersymmetric models with gauged fla-
vor symmetries. There are several sources of flavor violation in these theories. The
dominant flavor violation is the tree-level D-term contribution to scalar masses gen-
erated by flavor symmetry breaking. We present a new approach for suppressing this
phenomenologically dangerous effects by separating the flavor-breaking sector from
supersymmetry-breaking one. The separation can be achieved in geometrical setups
or in a dynamical way. We also point out that radiative corrections from the gaugi-
nos of gauged flavor symmetries give sizable generation-dependent masses of scalars.
The gaugino mass effects are generic and not suppressed even if the dominant D-term
contribution is suppressed. We also analyze the constraints on the flavor symmetry
sector from these flavor-violating corrections.
Supersymmetric extension of the standard model (SM) has been found to be very at-
tractive, in particular, as a solution to the hierarchy problem. Superpartners of the SM
fields are expected to be detected in future experiments. Even at present, supersymmetry
(SUSY) breaking parameters are constrained from flavor-changing neutral current (FCNC)
processes as well as CP violation . That is the so-called SUSY flavor problem and re-
quires sfermion masses between the first and second generations being degenerate, unless
they are sufficiently heavy or fermion and sfermion mass matrices are aligned quite well.
Such requirements for sfermion masses have been tried to be realized by considering flavor-
blind SUSY breaking and/or mediation mechanisms. Actually, various types of flavor-blind
mechanisms have been proposed in the literature -.
Understanding the origin of fermion masses and mixing angles is also one of the im-
portant issues in particle physics. Three copies of SM generations have the exactly same
quantum numbers except for their masses, i.e. Yukawa couplings to the Higgs field. In
the SM, the exact flavor universality is violated only in the Yukawa sector. One expects
that the hierarchical structure of Yukawa matrices is explained by some dynamics beyond
the SM, and such additional dynamics necessarily leave some imprint of flavor violation.
In supersymmetric models, a mechanism for realistic fermion masses breaks the flavor
universality and generally makes the corresponding sfermion masses flavor-dependent.
One of the salient mechanisms for generating hierarchical Yukawa couplings is the
Froggatt-Nielsen (FN) mechanism with additional U(1) gauge symmetries [7, 8]. In the
FN scenarios, flavor-dependent U(1) charges are assigned to matter fields so that realistic
Yukawa matrices are effectively realized in terms of higher-dimensional operators. There
is a certain reason to believe that the U(1) symmetries should be gauged; any global sym-
metry is expected to be unstable against quantum gravity and hence accidental. Therefore
the U(1) flavor symmetries that exactly control such operators should be gauged. Conse-
quently, after flavor symmetry breaking, the auxiliary D fields of the U(1) vector multiplets
give additional contribution to sfermion masses, which is proportional to U(1) charges and
hence flavor-dependent [9, 10]. That indeed gives significant modification of sparticle spec-
trum and in some cases is confronted with the SUSY flavor problem. This is called the D
problem in the present paper.
In this paper, we present an idea for suppressing the flavor-dependent D-term contri-
bution to sfermion masses. The D-term contribution is generated when one integrates
out heavy fields which develop vacuum expectation values (VEVs) that break the U(1)
gauge symmetries. As will be reviewed below, the modification of low-energy spectrum is
determined by soft SUSY-breaking masses of the heavy fields. Our idea is that the D-term
contribution is suppressed if the soft masses of the heavy fields can be made small. We will
present illustrative models where this idea is realized in a dynamical or geometrical way.
Unlike other approaches, the models presented here have an advantage that origins of the
Yukawa hierarchy can be addressed within the same framework.
However, even if the dominant D-term contribution is suppressed, there still remains
a flavor-violating effect from the U(1) gauge symmetries; flavor-dependent scalar masses
are radiatively generated by U(1) gaugino loop graphs, which involve the gaugino-fermion-
sfermion vertices proportional to U(1) quantum numbers. Such radiative effect is described
in terms of renormalization-group equations (RGEs) for scalar masses. Unlike the above
tree-level D-term contribution, the gaugino loop effect does not depend on SUSY-breaking
scalar masses. Therefore even with reduced D-term contribution, e.g. by assuming the
universal sparticle spectrum, flavor violation from the U(1) flavor symmetries still remains
and turns to be detectable signatures for flavor physics.
This paper is organized as follows. In Section 2 we describe scalar masses in super-
symmetric theories with a gauged abelian flavor symmetry. In addition to the D-term
contribution, we point out subleading but sizable flavor violation due to the abelian gaug-
ino. In Section 3, we will discuss how to suppress the dominant D-term flavor violation.
After a brief survey of the existing proposals in Section 3.1, we present in Section 3.2 the
models with extra dimensions, while another model based on four-dimensional supercon-
formal dynamics is presented in Section 3.3. We also examine how much degeneracy of
sfermion masses is expected in these models. In Section 4, radiative corrections from the
soft gaugino mass are shown to be potentially dangerous and give significant constraints
on model parameters such as soft gaugino masses. Section 5 summarizes our results.
2 Sfermion masses with U(1) gauge symmetry
In this section, we discuss sfermion masses in the presence of a U(1) horizontal gauge
symmetry, denoted by U(1)Xthroughout this paper. The D-term effect has been considered
in various contexts .As simple examples, we study two types of models where a
non-vanishing VEV of the auxiliary component D of U(1)X vector multiplet is actually
generated. However the properties presented here are generic for any model of D-term
As the first example, let us consider a pseudo-anomalous abelian gauge symmetry, which
often appears in string models . In this case, the Fayet-Iliopoulos (FI) term is generated
by a non-vanishing VEV of the dilaton or moduli field, whose nonlinear shift cancels the
U(1)Xanomaly. We treat the coefficient of the FI term as a constant, for simplicity.∗The
supersymmetric scalar potential relevant to the U(1)Xgauge sector is written as
VSUSY = −
+ ···. (2.1)
∗If the dilaton and moduli fields are treated as dynamical fields, the FI term and K¨ ahler metric of the
FN field depend on these fields. In this case, as was shown in Ref. , the formula for D-term contribution
becomes different from that in softly-broken global SUSY models with a constant FI term. The suppression
mechanism presented in this paper should be reconsidered in such a case.
Here gX is the gauge coupling and ξFI is the coefficient of the FI term, which we take
positive without lose of generality. The U(1)Xcharges of scalars φiand¯φiare denoted as
qi(> 0) and ¯ qi(< 0), respectively. The equation of motion for D is
¯ qj|¯φj|2. (2.2)
Then in the supersymmetric limit D = 0, the negatively-charged fields¯φigenerally develop
nonzero VEVs and the abelian gauge symmetry is broken at MX∼ ξ1/2
with vanishing VEVs remain massless but¯φidecouple around the MXscale.
In this work, we assume that the U(1)X breaking scale MX ∼ ξ1/2
cutoff Λ of the theory. It is known [7, 8] that the ratio ξ1/2
hierarchy of Yukawa couplings. Consider, for example, a superpotential W includes the
following non-renormalizable operators
FI. The scalar fields
FI is smaller than a
FI/Λ can be an origin of the
W = yij
where H denotes the electroweak Higgs field, and¯φ is a negatively-charged field that
develops a nonzero VEV of order ξ1/2
conservation to be nij= (qi+qj+qH)/|¯ q|. The operators (2.3) induce the effective Yukawa
FI. The power nijis determined by the U(1)Xcharge
ij= λnijyij,λ ≡
The factor λ represents a unit of hierarchy of Yukawa couplings and is usually taken to be
of the order of the Cabibbo angle. In this way, a realistic hierarchy of low-energy Yukawa
couplings can be obtained by assigning different charges to matter fields φ [7, 8].
We add a remark on possible realization of the FN mechanism in weakly-coupled heterotic
string models. In this case, the U(1) anomaly is cancelled by a nonlinear shift of the
dilaton-axion multiplet , and the FI term ξFIis generated at loop level. Consequently
ξFInaturally has an appropriate size for the Yukawa hierarchy. This possibility provides
us with a strong motivation for regarding an anomalous U(1) as a gauged flavor symmetry.
Note also that in the case of anomalous U(1) symmetry, the axion-gauge mixing generates
an additional contribution to the gauge boson mass, 2(ξFI/Λ)2. Since this contribution is
suppressed by ξFI/Λ2compared with that from the scalar VEV ?¯φ?, we will neglect this
effect in our analysis below.
The SUSY vacuum is shifted when soft SUSY-breaking masses for scalars are introduced.
The scalar potential is now given by
V = VSUSY+
terms irrelevant to our analysis. In VSUSY, the D component is replaced with scalar fields
iand ¯ m2
jare arbitrary mass parameters. The ellipsis denotes other SUSY-breaking
through the equation of motion (2.2). Minimizing the potential with respect to negatively-
charged fields, we find that the D component obtains a VEV
In this expression, ¯ m2/|¯ q| is the minimum value of ¯ m2
tive potential around the minimum along the¯φidirection takes a value of O(¯ m2
Without superpotential, only the scalar fields with such minimal value of ratio, ¯ m2/|¯ q|,
can contribute to the above equation. For example, in the case of universal scalar masses,
(a combination of) scalar fields with the largest negative value of U(1) charge obtain the
VEV (2.6). In this way, the D-flat direction is lifted by SUSY-breaking masses of scalar
fields with negative quantum numbers. From Eqs. (2.1) and (2.6), one finds a formula for
the D-term contribution to the masses of light scalars:
i/|¯ qi| in the model, since the effec-
Di= qi?D? =
|¯ q|¯ m2. (2.7)
The most important property of the formula (2.7) is that the induced scalar masses
squared are proportional to their U(1)Xcharges. This fact gives a significant implication
to flavor physics. As explained above, realistic low-energy Yukawa couplings are generated
by assigning different charges to matter fields. If all SUSY-breaking masses are of the
same order of magnitude, this scalar mass difference leads to large FCNC amplitudes. For
example, lepton-flavor violation from flavor symmetry D-terms was discussed in .
Besides the flavor problems, the induced scalar masses (2.7) have several interesting
properties. Firstly, the contribution is independent of the U(1)Xgauge coupling constant
gX. Therefore the formula (2.7) remains valid even if gXis small. [Note, however, that the
complete global limit (gX→ 0) cannot be taken since the U(1)Xsymmetry is broken only
if the condition g2
around the origin of the moduli space.] Secondly, the D-term contribution does not depend
on the symmetry-breaking scale MX, either. The D-term contribution is proportional to
a tiny deviation from supersymmetric conditions (flat directions), and the deviation is de-
termined by supersymmetry breaking independently of the gauge symmetry breaking .
As a result of these properties, the scalar masses induced via the D term appear for any
value of gauge coupling and symmetry-breaking scale.†It is also found from (2.7) that the
normalization of U(1)Xcharges does not affect the size of D-term contribution.
The D-term contribution also appears for non-anomalous abelian gauge symmetries.
Here we consider a model which contains vector-like fields Y and¯Y with U(1)X charges
±qY, and a gauge singlet Z. Taking a renormalizable and gauge-invariant superpotential
W = fZ(Y¯Y − M2
†This is true provided that soft SUSY-breaking terms are present already at the scale of scalar VEVs.
On the other hand, it will be intuitively clear that the D-term contribution is absent if soft masses arise
only at low energy as in the gauge mediation of SUSY breaking . This can be seen from the fact that,
to correctly determine the scalar VEVs, one has to use the renormalization-group improved potential 
in which all running parameters are evaluated at the VEV scale.
X> ¯ m2/(−¯ qξFI) is satisfied so that the scalar potential (2.5) is unstable