Large hierarchies from approximate R symmetries.

Abstract

We show that hierarchically small vacuum expectation values of the superpotential in supersymmetric theories can be a consequence of an approximate R symmetry. We briefly discuss the role of such small constants in moduli stabilization and understanding the huge hierarchy between the Planck and electroweak scales.

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arXiv:0812.2120v1 [hep-th] 11 Dec 2008
TUM-HEP-705/08; DESY 08-189; LMU-ASC 60/08
Large hierarchies from approximate R symmetries
Rolf Kappl1, Hans Peter Nilles2, Sa´ ul Ramos-S´ anchez3, Michael Ratz1,
Kai Schmidt-Hoberg1, Patrick K. S. Vaudrevange4
1Physik Department T30, Technische Universit¨ at M¨ unchen,
James-Franck-Strasse, 85748 Garching, Germany
2Bethe Center for Theoretical Physics and Physikalisches Institut der Universit¨ at Bonn,
Nussallee 12, 53115 Bonn, Germany
3Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22603 Hamburg, Germany
4Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universit¨ at M¨ unchen, 80333 M¨ unchen, Germany
We show that hierarchically small vacuum expectation values of the superpotential in supersym-
metric theories can be a consequence of an approximate R symmetry. We briefly discuss the role
of such small constants in moduli stabilization and understanding the huge hierarchy between the
Planck and electroweak scales.
I.INTRODUCTION
One of the major puzzles in contemporary physics is
the existence of large hierarchies in nature, such as the ra-
tio between the Planck and electroweakscales MP/mW∼
1017. Some of the most promising explanations of such
hierarchies rely on dimensional transmutation. Here the
dynamical scale Λ = MPe−a/g2(with g and a denot-
ing the gauge coupling and a constant, respectively) can
be naturally much smaller than the fundamental scale.
However, if one is to embed this mechanism in a more
fundamental framework, one often encounters the prob-
lem that there has to be a hierarchically small quantity
right from the start. Concretely, if one is to make use
of the dynamical scale in string theory, one has first to
fix the modulus that determines the coupling strength.
This in turn often requires the introduction of a small
constant. One faces then the well-known “chicken-or-egg
problem”.
Motivated by results obtained in the framework of
string theory model building, we present here a poten-
tial resolution of the problem. We shall show that, if
the superpotential in a supersymmetric theory exhibits
an approximate U(1)Rsymmetry, it generically acquires
a suppressed vacuum expectation value (VEV). Such ac-
cidental U(1)R symmetries which get broken at higher
orders are naturally present in string compactifications.
They arise as remnants from exact, discrete R symme-
tries. Such symmetries allow us to control the VEV of
the (perturbative) superpotential and, in particular, to
avoid deep anti-de Sitter vacua. We will discuss the role
of the resulting hierarchically small superpotential VEVs
in the context of moduli stabilization in string theory, for
giving a plausible explanation of the huge hierarchy be-
tween MPand mW, and for providing, in the context of
a class of promising string models [1], a solution to the µ
problem of the minimal supersymmetric standard model
(MSSM).
II.
AS A CONSEQUENCE OF A U(1)R SYMMETRY
SUPERSYMMETRIC MINKOWSKI VACUA
Consider a superpotential of the form
W =
?
cn1···nMφn1
1···φnM
M. (1)
Here and in the following we work in Planck units, i.e.
we set MP= 1 unless stated differently. Assume that W
has an exact R symmetry, under which W has R charge
2,
W → e2i αW ,(2)
and the fields transform as
φj → φ′
j= eirjαφj
(3)
such that each monomial in (1) has total R charge 2.
Let ?φi? denote a field configuration which solves the
F-term equations,
Fi =
∂W
∂φi
= 0 at φj= ?φj? ∀ i,j . (4)
Consider now an infinitesimal U(1)Rtransformation,
W (φi) → W (φ′
i) = W (φi) +
?
j
∂W
∂φj∆φj. (5)
At φj = ?φj? the superpotential goes into itself, which
can only be consistent with (2) if W = 0 at φj = ?φj?.
This proves that, if the F equations are satisfied, W van-
ishes.
A few comments are in order. First, this statement
holds regardless of whether the configuration ?φi? pre-
serves U(1)Ror breaks it spontaneously. Second, in the
context of supergravity, the statements above imply that
the DiW vanish for φi = ?φi?, i.e. also the supergrav-
ity F terms vanish and one obtains a supersymmetric

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Minkowski vacuum. Third, our findings are related to an
observation by Nelson and Seiberg made in [2], where it is
stated that, in order to have a theory without supersym-
metric ground state, the superpotential has to exhibit a
continuous R symmetry. The statements do, however,
not tell us whether or not a theory with a superpotential
exhibiting a continuous R symmetry has a supersymmet-
ric ground state or not. Our findings and [2] imply that, if
there is a continuous R symmetry, there are two options:
1. there is a supersymmetric ground state with W = 0
(with U(1)Rspontaneously broken or unbroken);
2. there is no supersymmetric ground state, and in the
ground state U(1)Ris spontaneously broken [2].
In this letter we focus on case 1. If the U(1) that acts
on the scalar components of the superfields gets spon-
taneously broken at φi = ?φi? (which is the case if, for
instance, all ?φi? are non-trivial), it follows then from
Goldstone’s theorem that there is a massless mode, the
so-called R axion.
III.SMALL CONSTANTS FROM
APPROXIMATE U(1)R SYMMETRIES
Let us now study what happens if the R symmetry is
‘slightly’ broken, i.e. by higher order terms. We can write
the superpotential as
W (φi) = W0(φi) +
?
j
Wj(φi) , (6)
where W0(φi) consists of monomials up to order N − 1
which preserve the R symmetry while the Wj(φi) are
monomials of order ≥ N which break the R symme-
try. This means that the superpotential transforms under
U(1)Ras
W (φi) → e2iαW0(φi) +
?

j
eiαRjWj(φi)
≃ W (φi) + iα
2W0(φi) +
?
j
RjWj(φi)

(7)
with Rj?= 2, and
W (φi) → W (eiα riφi)
≃ W (φi) + iα
?
j
∂W
∂φjrjφj. (8)
Combining these two expressions and assuming that the
F-terms vanish in our vacuum,∂W
∂φi= 0, we see that
?W (φi)? = −1
2
?
j
(Rj− 2)?Wj(φi)? .(9)
This means that in the case of an approximate U(1)R
symmetry one obtains suppressed superpotential VEVs,
written symbolically as
?W ? ∼ ?φ?≥N. (10)
In many situations there is a mild hierarchy between
the fundamental scale and a typical VEV, ?φ?/MP< 1.
This is, for instance, the case in string models where a
U(1) factor appears ‘anomalous’, and where the one-loop
Fayet-Iliopoulos term forces some VEVs to be roughly
one order of magnitude smaller than MP [3]. Accord-
ing to the above discussion, the suppression of ?W ? gets
then enhanced by the Nthpower of this mild hierarchy,
similarly to the Froggatt-Nielsen picture [4].
Further, we have seen that there might be a Goldstone
mode η. With explicit U(1)Rbreaking, it will generically
receive a mass, mη∼ ?φ?≥N−2. (The “−2” comes from
the second derivative.) In supergravity theories, ?W ? sets
the gravitino mass,
m3/2 ≃ ?W ? .(11)
This leads then to the expectation that there is a mode
whose (supersymmetric) mass scales like m3/2,
mη ∼
m3/2
?φ?2. (12)
Let us comment that, if one is to include supergravity
effects, W ?= 0 does not necessarily imply anti-de Sitter
solutions (see e.g. the discussion in [5, section 4]).
IV.EXPLICIT STRING THEORY
REALIZATION
One of the central themes of string theory is the is-
sue of moduli stabilization, which is closely connected
to the question of supersymmetry breaking. In the tra-
ditional approach, supersymmetry is broken by dimen-
sional transmutation [6], e.g. by gaugino condensation [7].
However, for this elegant mechanism to work, one needs
first to fix the gauge coupling, whose strength is given by
the VEV of the dilaton S or another modulus in string
theory. This can be achieved in various ways: for in-
stance, in the race-track scheme [8] one has two com-
peting non-perturbative superpotentials which provide a
non-trivial minimum of the dilaton potential. The draw-
back of this mechanism is that it only works if one has
two rather large ‘hidden’ gauge groups with rather spe-
cial matter contents. A somewhat more economic scheme
is that of K¨ ahler stabilization [9, 10] where one needs
only one hidden sector. However, in the relevant regime
where dilaton stabilization may be achieved the theory
is not calculable. More recently, an alternative has been
studied (with the most prominent example being that of
KKLT [11]) where the superpotential is of the form
W = c + Ae−a S. (13)

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The first term c is a constant and the second term reflects
hidden sector strong dynamics, i.e. S is related to the
gauge coupling, ReS ∝ 1/g2, and a is related to the β-
function of the hidden gauge group. In the KKLT setup,
the constant comes from fluxes. The minimum of the
scalar potential for S occurs at a point where
|aS Ae−aS| ∼ |c| . (14)
The VEV of W , i.e. the gravitino mass, is of the same
order. In order to have MSSM superpartner masses at the
TeV scale, the gravitino mass cannot exceed O(100)TeV,
hence
|c| ? 10−12
(15)
in Planck units. The small scale in this setting is there-
fore not explained by dimensional transmutation but
originates from the smallness of the constant c. KKLT
and others argue that, due to the large number of vacua,
some of them might have such c by accident.
In what follows, we will exploit the observation of sec-
tion III that small VEVs of the (perturbative) superpo-
tential can be explained by an approximate U(1)Rsym-
metry. We will use this in order to discuss moduli sta-
bilization in the context of the heterotic string. We fo-
cus on orbifold compactifications [12] since they possess
many (and well-understood) discrete symmetries, which,
as it turns out, imply approximate U(1)Rsymmetries of
the superpotentials describing the effective field theories
derived from these constructions. As we shall see, super-
potential VEVs of the order 10−O(10)can naturally be
obtained. Orbifold compactifications allow us to embed
the MSSM into string theory [14,13,1].
In our calculations we focus on the models of the ‘het-
erotic MiniLandscape’ [15,1]. These models exhibit the
standard model gauge group and the chiral matter con-
tent of the MSSM. They are based on the
with three factorizable tori (see [16,13] for details). The
discrete symmetry of the geometry leads to a large num-
ber of discrete symmetries governing the couplings of the
effective field theory [17, 18] (cf. also [16,13,19]). Apart
from various bosonic discrete symmetries, one has a
?6-II orbifold
[?6×
?3×
?2]R
(16)
symmetry; other orbifolds have similar discrete symme-
tries. Further, in almost all of the MiniLandscape models
there is, at one-loop, a Fayet-Iliopoulos (FI) D-term,
VD ⊃ g2
??
i
qi|φi|2+ ξ
?2
, (17)
where the qi denote the charges under the so-called
‘anomalous U(1)’. It turns out that, in all models with
non-vanishing FI term, ξ is of order 0.1 (see [13] for an
explicit example). The first step of our analysis is to iden-
tify a set of standard model singlets φiwith the following
properties:
• giving VEVs to the φi allows us to cancel the FI
term;
• there is no other field that is singlet under the gauge
symmetries left unbroken by the φiVEVs.
These properties ensure that the ?φi? can be consistent
with a vanishing D-term potential and that the F-terms
of all other massless modes vanish, implying that it is suf-
ficient to derive the superpotential terms involving only
the φifields. A crucial property of these superpotentials
is that they exhibit accidental U(1)Rsymmetries that get
only broken at rather high orders N. As discussed, this
can be regarded as a consequence of high-power discrete
R symmetries (equation (16)). N depends on the cho-
sen φi configuration; as a general rule we find that the
more φifields are considered, the lower N values emerge.
For instance, in a model where only seven fields are con-
sidered, we obtain N = 26, on the other hand, in the
model 1 of [1] with 24 fields switched on, U(1)Rgets bro-
ken at order 9.
Given non-trivial solutions to the F-term equations,
φi∂W
∂φi
= 0 ,with φi?= 0 , (18)
one can use complexified gauge transformations to en-
sure vanishing D-terms as well [20]. Although D-term
constraints do not fix the scale of the ?φi? in general, the
requirement to cancel the FI term introduces the scale
√ξ ∼ 0.3 into the problem. We search for solutions of
VD= VF = 0 in the regime |φi| < 1, and find that they
exist. We explicitly verify that for such solutions the su-
perpotential is hierarchically small, ?W ? ∼ ?φ?N, where
?φ? denotes the typical size of a VEV. A very important
property of many of these configurations is that all fields
acquire (supersymmetric) masses. Hereby typically only
one field has a mass of the order mη(see equation (12))
while the others are much heavier. We have also checked
that these features are robust under supergravity correc-
tions.
Altogether we find that in the models under consider-
ation one obtains isolated supersymmetric field configu-
rations with |φi| < 1 where the VEV of the perturbative
superpotential ?W ? is hierarchically small.
Before discussing applications, let us compare our
findings to other recent results [21]. There, using the
stringy selection rules, so-called ‘maximal vacua’ were
constructed in which the superpotential vanishes term
by term (and to all orders). In our approach, each su-
perpotential term composed out of φi fields acquires a
non-trivial VEV, but to the order at which the accidental
U(1)Ris exact, all terms cancel non-trivially. At higher
orders, a non-trivial VEV of W gets induced.
Let us now briefly sketch how this can be used in order
to stabilize the dilaton, whose VEV determines the gauge
coupling. After integrating out the φi fields, one is left
with a superpotential of the form (13),
Weff = c + Ae−aS,(19)

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where c = ?W ? = 10−O(10), and Ae−aSdescribes some
non-perturbative dynamics, such as gaugino condensa-
tion [7,22,23,24]. As we have discussed before in equa-
tion (14), this superpotential leads to a non-trivial mini-
mum for the dilaton. In the MiniLandscape models, real-
istic gauge couplings are correlated with favorable sizes of
the dynamical scale, Ae−a S/M2
typical expectation values ?W ? = 10−O(10)one obtains
reasonable gauge couplings. The fixing of the T-moduli
and other issues such as ‘uplifting’ will be studied else-
where.
Another application of our findings concerns the µ
term of the MSSM. In [26] it has been proposed that
in models in which the field combination huhd(with hu
and hddenoting the up-type and down-type Higgs fields,
respectively) is completely neutral w.r.t. all symmetries
there is an interesting relation between the Higgs mass
coefficient µ and ?W ?,
µ ∼ ?W ? .
The heterotic MiniLandscape [15] contains many models
in which the Higgs pair (and only the Higgs pair) has
this property. Apart from the above property, such mod-
els exhibit ‘gauge-top unification’, i.e. the top Yukawa
coupling is of the order of the gauge coupling, as well as
many other desirable properties. In a concrete example,
the benchmark model 1A of [1], it was found that solving
the F-term equations for the superpotential up to order 6
always leads to ?W ? = 0. We have now obtained a better
understanding of this fact: there is a U(1)R symmetry
that holds up to order 11, explaining this property. It is
amazing to see that these models, constructed in order to
reproduce the MSSM spectrum and gauge interactions,
exhibit so many appealing properties automatically.
P∼ TeV [25]. Hence, for
(20)
V.CONCLUSIONS
We have shown that approximate U(1)R symmetries
can explain the appearance of hierarchically small
constants.We find that at configurations where the
F-term equations are solved, the superpotential goes
like ?W ? ∼ ?φ?Nwith ?φ? denoting a typical expectation
value and N being the order at which U(1)Rgets broken.
We have analyzed various heterotic orbifold models and
found that there, due to the presence of high-power
discrete R symmetries, approximate U(1)Rs are generic.
We have explicitly solved the F-term equations in several
models, thus obtaining points in field space in which the
F- and D-term potentials vanish, and confirmed that,
for |φi| < 1, the superpotential is hierarchically small.
We have argued that such suppressed superpotential
expectation values can be the origin for the appearance
of large hierarchies in nature:
the gravitino mass, which in schemes with low-energy
supersymmetry sets the weak scale, and can be used to
stabilize the string theory moduli at realistic values.
they fix the scale of
Acknowledgments.
F. Br¨ ummer,
useful discussions. This research was supported by the
DFG cluster of excellence Origin and Structure of the
Universe, the European Union 6th framework program
MRTN-CT-2006-035863 ”UniverseNet”, LMUExcellent
and the SFB-Transregios 27 ”Neutrinos and Beyond”
and 33 ”The Dark Universe” by Deutsche Forschungsge-
meinschaft (DFG).
Wewould like
J.
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