Noscale SU(5) superGUTs
Eur. Phys. J. C
John Ellis 1 2
Jason L. Evans 0
Natsumi Nagata 6
Dimitri V. Nanopoulos 3 4 5
Keith A. Olive 7
0 School of Physics , KIAS, Seoul 130722 , Korea
1 Theoretical Physics Department , CERN, 1211 Geneva 23 , Switzerland
2 Theoretical Physics and Cosmology Group, Department of Physics, King's College London , Strand, London WC2R 2LS , UK
3 Division of Natural Sciences, Academy of Athens , Athens 10679 , Greece
4 Astroparticle Physics Group, Houston Advanced Research Center (HARC) , Mitchell Campus, Woodlands, TX 77381 , USA
5 George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station , TX 77843 , USA
6 Department of Physics, University of Tokyo , Tokyo, Bunkyoku 1130033 , Japan
7 School of Physics and Astronomy, William I. Fine Theoretical Physics Institute, University of Minnesota , Minneapolis, MN 55455 , USA
We reconsider the minimal SU(5) grand unified theory (GUT) in the context of noscale supergravity inspired by string compactification scenarios, assuming that the soft supersymmetrybreaking parameters satisfy universality conditions at some input scale Min above the GUT scale MGUT. When setting up such a noscale superGUT model, special attention must be paid to avoiding the Scylla of rapid proton decay and the Charybdis of an excessive density of cold dark matter, while also having an acceptable mass for the Higgs boson. We do not find consistent solutions if none of the matter and Higgs fields are assigned to twisted chiral supermultiplets, even in the presence of GiudiceMasiero terms. However, consistent solutions may be found if at least one fiveplet of GUT Higgs fields is assigned to a twisted chiral supermultiplet, with a suitable choice of modular weights. Spinindependent dark matter scattering may be detectable in some of these consistent solutions. Globally supersymmetric grand unification has long been an attractive framework for unifying the nongravitational interactions, with the minimal option using the gauge group SU(5) [13]. When incorporating gravity, one must embed such a supersymmetric grand unified theory (GUT) within some supergravity theory, and an attractive option is noscale supergravity [47]. This has the advantages that it leads to an effective potential without holes of depth O(1) in natural units, and emerges in generic string compactifications [8]. Noscale supergravity also allows naturally for the possibility of Planckcompatible cosmological inflation [9, 10]. In

general, a noscale K?hler potential contains several
moduli Ti , but here we consider scenarios in which the relevant
dynamics is dominated by a single volume modulus field T .
The construction of noscale supergravity GUTs
encounters significant hurdles, such as fixing the
compactification moduli. Moreover, pure noscale boundary conditions
require that all the quadratic, bilinear and trilinear scalar
couplings m0, B0 and A0 vanish, leading to phenomenology
that is in contradiction with experimental constraints.
However, this issue may be avoided in models with (untwisted or
twisted) matter fields with nonvanishing modular weights
as we show below.
The simplest possibility for soft supersymmetry
breaking is to postulate universal values of m0, B0 and A0, as
in the constrained minimal supersymmetric Standard Model
(CMSSM) [11?38]. With the inclusion of a universal
gaugino mass, m1/2, the CMSSM is a fourparameter theory.1
Minimal supergravity places an additional boundary
condition, relating B0 and A0 ( B0 = A0 ? m0) making it a
threeparameter theory [39?41]. Noscale supergravity,
however, is effectively a oneparameter theory since we require
m0 = A0 = B0 = 0. Another oneparameter theory in this
context is pure gravity mediation [42?47], in which the
gaugino masses, A and B terms2 are determined by anomaly
mediation [49?53] leaving only the gravitino mass, m3/2 = m0 as
a free parameter.
These boundary conditions may be too restrictive if they
are imposed at the GUT scale, MGUT, defined as the
renormalization scale where the two electroweak gauge couplings
1 In addition, one must choose the sign of ?, which we take here to
positive.
2 In order to get electroweak symmetry breaking to work, the B terms in
these models also get a contribution from a Giudice?Masiero term [48].
are unified. There is, however, no intrinsic reason that the
boundary conditions for supersymmetry breaking coincide
with gauge coupling unification. Separating these two scales
opens the door for socalled subGUT models [37,38,54?56]
where the input universality scale differs from the GUT scale
with Min < MGUT or the possibility that the boundary
conditions are imposed at some higher input scale Min > MGUT,
a scenario we term superGUT [57,58].
However, the regions of parameter space with acceptable
relic density and Higgs mass typically require quite
special values of the GUT superpotential couplings and rather
large values of tan ? [59], and hence a proton lifetime that
is unacceptably short. In order to accommodate smaller
values of tan ? and hence an acceptably long proton lifetime,
we consider nonzero Giudice?Masiero (GM) terms [48] in
the K?hler potential. In this way we are able to avoid the
Scylla of rapid proton decay and the Charybdis of an
excessive density of cold dark matter, while also having an
acceptable value of the Higgs mass.3 Furthermore, when noscale
boundary conditions are applied at the GUT scale, the
lightest sparticle in the spectrum is typically a stau (or the stau
is tachyonic). Applying the boundary conditions above the
GUT scale as in a superGUT model can alleviate this
problem [62].
The outline of this paper is as follows. In Sect. 2, we
review our theoretical framework, with our setup of the
minimal supersymmetric SU(5) model described in Sect. 2.1,
our noscale supergravity framework inspired by string
compactification scenarios described in Sect. 2.2 and the vacuum
conditions and the relevant renormalizationgroup equations
(RGEs) set out in Sect. 2.3. We describe our key results in
Sect. 3. We explore in Sect. 3.1 scenarios in which none of
the matter and Higgs supermultiplets are twisted, and we
find no way to steer between Scylla and Charybdis with an
acceptable Higgs mass in this case. However, as we show
in Sect. 3.2, this is quite possible if one or the other (or
both) of the GUT fiveplet Higgs supermultiplets is twisted.
Spinindependent dark matter scattering may be observable
in some of the cases studied. Finally, Sect. 4 discusses our
results.
2 SuperGUT CMSSM models
2.1 Minimal supersymmetric SU(5)
The minimal supersymmetric SU(5) GUT [2,3], was recently
reviewed in [58] and we recall here the aspects most needed
for our discussion. The minimal renormalizable
superpotential for this model is given by
3 The Higgs boson mass and dark matter for noscale models with non
universal Higgs masses was also considered in [60,61].
where Greek sub and superscripts denote SU(5) indices, and
Eqi.s(1th)e, thtoetaaldlyjoiannttimsyumltimpleettric t?en?so2r wAitTh A,1w23h4e5re=the1.TIAn
( A = 1, . . . , 24) are the generators of SU(5) normalized so
that Tr(T A T B ) = ?AB /2, is responsible for breaking SU(5)
to the Standard Model (SM). The scalar components of
are assumed to have vevs of the form
= V ? diag (2, 2, 2, ?3, ?3) ,
where V ? 4? /? , causing the GUT gauge bosons X to
acquire masses MX = 5g5V , where g5 is the SU(5) gauge
coupling.
The multiplets H and H in Eq. (1) are 5 and 5
representations of SU(5), respectively, and contain the MSSM Higgs
fields. In order to realize doublettriplet mass splitting in the
H and H multiplets, we impose the finetuning condition
?H ? 3?V V . In this case, the colortriplet Higgs states
have masses MHC = 5?V , the masses of the color and weak
adjoint components of are M = 5? V /2, and the singlet
component of acquires a mass M 24 = ? V /2.
The multiplets i in Eq. (1) are 5 representations
containing the lefthanded SM matter fields Di and Li , and the i
are 10 representations of SU(5) containing the lefthanded
Qi , U i , and Ei , where the index i = 1, 2, 3 denotes the
generations.
The soft supersymmetrybreaking terms in the minimal
supersymmetric SU(5) GUT are
Lsoft = ?(m120)i j ?i?? j ? (m25)i j ?i?? j ? m2H H 2
? m2H H 2 ? m2 Tr( ? )
+ A5(h5)i j ?i?? ? j? H ?
1
+ B ? Tr 2 + 6 A? ? Tr 3 + BH ?H H H
where ?i and ?i are the scalar components of i and i ,
respectively, the ?A are the SU(5) gauginos. We use the
same symbols for the scalar components of the Higgs fields
as for the corresponding superfields. Moreover, we have
assumed that the Aterms are proportional to the
corresponding Yukawa couplings in the superpotential; we will see in
Sect. 2.2 that the Aterms arising in noscale supergravity
actually have this structure.
2.2 Noscale framework
We refer to [63] for a derivation of the soft terms arising in
noscale supergravity.4 Our startingpoint is a noscale K?hler
potential inspired by string compactification scenarios,
1
K = ?3 ln T + T? ? 3
which includes a volume modulus field, T , and both
untwisted and twisted matter fields, ?i and ?a , respectively,
the latter with modular weights na . We consider a generic
superpotential of the form
W = (T + c)? W2(?i ) + (T + c)? W3(?i )
where c is an arbitrary constant, and W2,3 denote bilinear and
trilinear terms with modular weights that are in general
nonzero and ? . When ?, ? = 0, the effective potential for T
is completely flat at the tree level, so it has an undetermined
vev, and the gravitino mass
varies with the value of this volume modulus.5 We assume
here that some Planckscale dynamics fixes T = T? = c, and
take c = 1/2 in the following.
In a standard noscale supergravity model with no twisted
fields and with weights ? = ? = 0, we would obtain
m0 = A0 = B0 = 0. However, in the scenario (5) soft terms
are induced, as were calculated in [63], which are
sectordependent:
?i : m0 = 0, B0 = ??m3/2,
?
A0 = 3m3/2 1 ? 3 ,
?
?a : m0 = m3/2, B0 = 2m3/2 1 ? 2
where we have assumed for simplicity that na = 0. This
universality renders the Aterms proportional to the Yukawa
couplings in Eq. (3). We also postulate in what follows
generalized Giudice?Masiero terms [48]
K = (cH (T + c)?H H H? + c (T + c)?
4 Related derivations of soft terms in string models with flux compact
ifications can be found in [64?66].
5 The parameter ? does not play any other role in our construction,
and its precise value is unimportant for our analysis.
If H , H? , and
? and B terms:
are untwisted, these induce corrections to the
= c m3/2,
If the fields are twisted, the shift in the ?terms is the same,
but the shift in B? is modified by ??H, cH, ? (2 ?
?H, )cH, [63]. Although the corrections to the B terms are
quite small, they are crucial for matching the GUT scale B
terms onto the MSSM B term at the GUT scale, as we see
below.
In the superGUT version of the CMSSM model we
impose the following universality conditions for the soft mass
parameters at a soft supersymmetrybreaking mass input
scale Min > MGUT:
(m120)i j = (m25)i j ? m20 ?i j ,
m H = m H = m
? m0,
A10 = A5 = A? = A? ? A0,
BH = B
? B0,
M5 ? m1/2,
with the input soft terms m0, A0 and B0 specified above.
In the above expressions, and in expressions throughout the
text, the B contribution is neglected since it is so small.
However, this contribution to the Bterms is included in all
calculations in order to satisfy the Bterm matching
condition.
We conclude this subsection by emphasizing that the
parameters introduced in Eqs. (4)?(6) and (9) above are
intrinsic to our stringinspired noscale supergravity
framework. The parameters na in (4) characterize the twisted
matter fields ?a , the parameters ?, ?, ? and ? in (5) are modular
weights that are unknown a priori, the value c of the volume
modulus is assumed to be fixed by some Planckscale
dynamics, and the parameters cH, and ?H, are needed to
characterize the Giudice?Masiero terms (9) that are a common
feature of supergravity models. Many such analogous
parameters would appear in any model based on N = 1
supergravity: here they are related to properties of an underlying
scenario, namely string compactification, which is necessarily
ambiguous at our present level of understanding. Our
subsequent phenomenological analysis may serve to give pointers
how these ambiguities could be reduced.
2.3 Vacuum conditions and renormalizationgroup equations
Since the Bterm boundary conditions are specified at Min,
we cannot use the Higgs minimization equations to determine
B and the MSSM ? term as is commonly done in the MSSM.
Instead, as in mSUGRA models, these conditions can be used
to determine ? and tan ? [40,41] as was done in the
noscale superGUT models considered in [59]. In [59], standard
noscale boundary conditions were used to identify regions
of parameter space with acceptable relic density and Higgs
mass. Typically, rather large values of tan ? were found and,
in addition, it was necessary to choose somewhat small values
of the coupling ? = O(0.01) with much larger values of ? =
O(1). All of these choices tend to decrease the proton lifetime
to unacceptably small values [58]. In order to reconcile the
proton lifetime with the relic density and Higgs mass, we
need to consider lower values of tan ? [38,58,67], which can
be accomplished when the GM terms (9) are included [46?
48,68].
The soft supersymmetry breaking parameters are evolved
down from Min to MGUT using the renormalizationgroup
equations (RGEs) of the minimal supersymmetric SU(5)
GUT, which can be found in [57,59,70?73], with
appropriate changes of notation. During the evolution, the GUT
couplings in Eq. (1) affect the running of the soft
supersymmetrybreaking parameters, which results in nonuniversality in the
soft parameters at MGUT. In particular, the GUT coupling
? contributes to the running of the Yukawa couplings, the
corresponding Aterms, and the Higgs soft masses. On the
other hand, ? affects directly only the running of ?, m , and
A? (besides ? and A? ), and thus can affect the MSSM soft
mass parameters only at higherloop level. Both ? and ?
contribute to the RGEs of the soft masses of matter multiplets
only at higherloop level, suppressing their effects on these
parameters.
At the unification scale MGUT (defined as the
renormalization scale where the two electroweak gauge couplings are
equal), the SU(5) GUT parameters are matched onto the
MSSM parameters. The matching conditions for the
Standard Model gauge and Yukawa couplings were discussed in
detail in [58]. The use of threshold corrections at the GUT
scale [74?76] allow us to determine the SU(5) gauge
coupling, g5, and the SU(5) Higgs adjoint vev, V , which in turn
allows us to fix the gauge and Higgs boson masses as
MX = 5g5V ,
which are inputs in the calculation of the proton lifetime.
As explained in [58], in order to allow both ? and ?
to remain as free parameters, we must include a
Plancksuppressed operator such as
6 For example, V = 9 ? 1016 ? 3 ? 1017 GeV in Fig. 1 for ? = 10?5.
where W ? T AW A denotes the superfields
corresponding to the field strengths of the SU(5) gauge vector bosons.
Such operators may make contributions comparable to other
threshold corrections when develops a vev [77?82]. We
have checked that the coefficient c5 takes reasonable values,
i.e., c5 < O(1).
There are other nonrenormalizable operators that should
be considered in any supergravitybased model. For example,
the operators of the form [83]
can have an important effect on the matching conditions for
the gauge couplings at the GUT scale as well. These operators
can split the masses of the SU(3)C and SU(2)L adjoint
components in , M 8 and M 3 , respectively. For ? ? O(1),
this nonrenormalizable operator gives a small splitting to the
masses of order V 2/MP M 3,8 . Although this splitting
is small, it still gives additional threshold corrections to the
gauge coupling matching conditions of order
where we have used M 8,3 25 ? V and M 3 ? M 8 =
V 2/MP for the last expression. In comparison to the
contribution coming from Eq. (15), which is of order 8V /MP , this
contribution is quite small.
In the case where ? is small, the mass splitting of M 3,8
becomes significant and ln(M 3 /M 8 ) is now order one. The
threshold correction to the gauge couplings coming from Eq.
(17) is now of order 1/(16? 2). However, for small ? , the
vev of grows and is now of order 1017 GeV.6 With a vev
this large, 8V /MP is much larger than a loop factor and we
can again safely neglect the contributions coming from the
operators in Eq. (17).
The matching conditions for the soft
supersymmetrybreaking terms were also discussed in detail in [58]. The
matching conditions for the gaugino masses [82,84] are given
by
We again find that the contribution of the dimensionfive
operator in Eq. (15) can be comparable to that of the oneloop
threshold corrections. MSSM soft masses and the Aterms
of the third generation sfermions, are given by
(26) to have the correct value at the GUT scale. As noted
earlier, this often leads to relatively large values of tan ? and
unacceptable low values for the proton lifetime.
Alternatively, we can introduce a GM term in the K?hler
potential as in Eq. (9). For now, we assume that all fields are
untwisted with weight ? = ?1. The shift in the ?terms is
O(MSUSY) and is irrelevant to the matching condition (22).
Similarly the shifts in most of the terms in (23) are of order
m23/2/MGUT and are much smaller than O(MSUSY).
However, there is a shift in
cH ?
Although this shift is also small, is multiplied by V /? in
(23), so that the overall shift in B is O(MSUSY). Thus the
shift in (23) becomes
m2Q = mU2 = m2E = m210, m2D = m2L = m2,
5
m22 ? m2Hu = m2H , m21 ? m2Hd = m2 ,
H
At = A10,
B = BH +
) ? m2 ],
The amount of finetuning required to obtain values of ?
and B that are O(MSUSY) is determined by these last two
equations. From Eq. (22), we find that we need to tune
?H ? 3?V  to be O(MSUSY). From Eq. (23), V /? should
be O(MSUSY), which requires   ? O(MS2USY/MGUT).
In standard noscale supergravity, = 0 and this is stable
against radiative corrections, as shown in Ref. [86]. As
discussed in Ref. [58], in order for Eq. (23) to have a real
solution for B , the condition A2? 8m2 should be satisfied
for ? ?. We have checked that this condition is always
satisfied over the parameter space we consider in Sect. 3.
The MSSM ? and B parameters can be determined by
using the electroweak vacuum conditions:
where B and (?1,2) denote loop corrections [87?89]. These
are run up to the GUT scale where the conditions (22) and
(23) are applied. However, in standard noscale supergravity,
the righthand side of (23) is determined by running down
the A and Bterms set by A0 = B0 = 0 (and similarly for
m2 ). Thus, (23) is not satisfied in general. Nevertheless, it
is often possible to find a value of tan ? that adjusts B? via
up to O(MSUSY/MGUT) corrections. This is now of
comparable size to other terms in (23), which can be satisfied for
any tan ?. The matching condition (23), therefore determines
a linear combination of the two GM terms.
Our noscale superGUT model is therefore specified by
the following set of input parameters:
where the trilinear superpotential Higgs couplings, ? and ? ,
are specified at Q = MGUT.
In the following we assume initially that all fields are
untwisted, so that m0 = 0, and assume vanishing modular
weights ? = ? = 0, so that A0 = B0 = 0. Later we consider
the effects of twisting one or both of the Higgs 5plets and
turning on the trilinear weight ? in order to allow nonzero
A0.
3 Results
3.1 Standard noscale supergravity with a GM term
It is well known that the CMSSM with noscale boundary
conditions is not viable. With m0 = A0 = B0 = 0, the
particle spectrum almost inevitably contains either a stau lightest
supersymmetric particle (LSP) or tachyonic stau. However,
this problem can be alleviated if the universal boundary
conditions are applied above the GUT scale [62]. In this case, the
running from Min to MGUT produces nonzero soft terms that
may be sufficiently large to produce a reasonable spectrum.7
7 Similar conclusions were reached in gauginomediated models in [90,
91].
?40
n
a
t
? = 1, ?' = 0.00001, ? > 0
125 111222556112265 125111112222125557265112257121527M121117222167726in =11122257711112220556111226581217G25111122221155572e26511V2257112257
Fig. 1 Sample noscale superGUT (m1/2, tan ?) planes for Min =
1018 GeV. In the left panel ? = ?0.1 and ? = 2, whereas in the right
panel ? = 1 and ? = 10?5. The brown shaded region has a stau LSP.
The regions compatible with the relic density determined by Planck and
other experiments are shaded dark blue, and the red dotdashed curves
are contours of constant Higgs mass as calculated using FeynHiggs,
which does not give stable results in the upper right portions of the
panels. In the left panel, the green curves are contours of c (m3/2/m1/2)2
and the proton lifetime is too short throughout. In the right panel, the
solid black contours show the proton lifetime in units of 1035 years,
which is acceptably long below the contour labeled 0.066. However,
the relic density is too large throughout this region
The basic noscale superGUT model was studied in detail
in [59]. There it was found that, for sufficiently large Min,
not only could a reasonable mass spectrum be obtained, but
also regions of parameter space with the correct relic
density and Higgs mass were identified. This region was further
explored in [92], with the aim of studying possible departures
from minimal flavor violation. There, for example, a
particular benchmark point was chosen with M5 = 1500 GeV,
Min = 1018 GeV, ? = ?0.1, ? = 2, which required
tan ? ? 52 as no GM term was included. One concern for this
benchmark is the proton decay rate, which is enhanced by the
combination of large tan ? and small ? (which induced a low
value for the Higgs colortriplet mass). Indeed, as we show
below, the proton lifetime is far too small in this minimal
SU(5) construction.
We show in Fig. 1 two examples of (m1/2, tan ?) planes
for fixed Min = 1018 GeV. In the left panel, we have chosen
? = ?0.1 and ? = 2. In the dark blue shaded strip, the
neutralino LSP relic density agrees with the value determined
by Planck and other experiments. To its left, in the brown
shaded region the stau is either the LSP or tachyonic. The
red dotdashed contours show the value of the Higgs mass
as computed using the FeynHiggs code [93].8 As one can
see, there is a region at large tan ? ? 52?55 for m1/2 ? 1?
8 Note that here we use FeynHiggs version 2.11.3, which gives a
slightly lower value of mh than the version used in [59]. Inaddition,
1.5 TeV that corresponds to the preferred region found in
[59].9 In this region the Higgs mass ? 122?124 GeV, which is
acceptable given the uncertainty in the mass calculated using
FeynHiggs. By including a GM term, we are able to probe
lower values of tan ? for the same set of input parameters.
Unfortunately, the proton lifetime is much too small over the
entire left panel, with a value of only 1025 years in the upper
left corner. We also show (in green) the contours of the GM
term. In this case, since ? ? , we assume cH = 0 and
show the contours of c (m3/2/m1/2)2.10 As one can see, the
contour for c = 0 runs through the region of good relic
density and Higgs mass found in [59].
In the right panel of Fig. 1 we show a similar plane but with
different choices of (?, ? ) = (1, 10?5), which are more
typical of the values required in [58]. In this case, with ? ? ,
the value of c (m3/2/m1/2)2 is very near ?0.25 all across
the plane. As long as cH is relatively small, one can see from
Eq. (28) that the value of cH has little effect on our estimates
Footnote 8 continued
since FeynHiggs does not produce stable results in the upper right
portion of the plane, the Higgs contours terminate in this region.
9 The slight differences between these and past results arise mostly
because here we do not force the strong gauge coupling to be equal to
the electroweak couplings at the GUT scale.
10 We make no specific assumption as regards the magnitude of m3/2,
except that it is large enough for the LSP to be the lightest neutralino,
rather than the gravitino.
of c (m3/2/m1/2)2, which are quoted assuming cH = 0.
The large ratio of ?/? is beneficial for increasing the
proton lifetime, and contours showing the lifetime are seen as
solid black curves in the lower right portion of the panel,
labeled in units of 1035 years;11 as the current
experimental limit is ? ( p ? K +?) > 6.6 ? 1033 years [97, 98], the
region with acceptable proton stability lies below the contour
labeled 0.066. Whilst it is encouraging that some region of
parameter space exists with a sufficiently long proton
lifetime and acceptable Higgs mass, the relic density is far too
large in this region: h2 ? O(100). Further exploration in
the (Min, ?, ? ) parameter space does not yield better results.
The Higgs mass can be made compatible with either the relic
density or the proton lifetime, but not both.
The left panel of Fig. 1 shows that, at fixed m1/2, the
value of mh decreases rapidly when tan ? 10. On the other
hand, the right panel of Fig. 1 shows that the proton lifetime
is unacceptably short for tan ? 10. As we discuss below
with several examples, these two problems can be avoided
simultaneously when tan ? = 7, for suitable choices of the
other superGUT model parameters Min, ? and ? . We do
not discuss in the following possible variations in the value
of tan ?, but have checked that values differing from 7 by
factors 2 are typically excluded by either mh or the proton
lifetime.
3.2 Twisted H and H Higgs fields
In this subsection we consider departures from the minimal
model discussed above that allow for more successful
phenomenology. We start by considering the consequences of a
twisted Higgs sector. As discussed above, tan ? must be
relatively low to obtain sufficiently long proton lifetimes.
However, in order to obtain a sufficiently large Higgs mass, tan ?
should not be too low. Choosing tan ? = 7 with ? = 10?5
optimizes both mh and ? p, so we fix those values for now. In
the following, we take ? = 0.6 and 1.
The superpotential (5) does not cover the case where
twisted fields couple to untwisted fields. If the Higgs 5plets
are twisted, then W3 contains Yukawa couplings between
the twisted Higgses and untwisted matter fields. In addition,
if remains untwisted, then W3 also contains a term
coupling one untwisted field ( ) and the twisted Higgs fields.
We define weights for each of the terms in W3: ?t , ?b, ??,
and ?? corresponding to the top and bottom Yukawa
couplings, the coupling of the Higgs adjoint to the 5plets, and
the adjoint trilinear, respectively. Similarly, we define
sepa11 Details of the calculation of proton decay rates can be found in
Refs. [38,58,67,94,95]. Here, we have take the phases in the GUT
Yukawa couplings [96] such that the proton decay rate is minimized
[58], which gives a conservative constraint on the model parameter
space.
rate weights ?H and ? for the two bilinears in W2. When
both H and H are twisted, A and B terms are given at the
input renormalization scale by
The Higgs soft squared masses are given by m23/2 in addition
to the usual supersymmetric contribution from ? (properly
shifted by the GM term).
We consider first the case where both Higgs 5plets are
twisted, and therefore receive equal soft
supersymmetrybreaking masses, m1 = m2 = m3/2. We start by taking
all of the modular weights ? = ? = 0 as before. Now,
however, there are nonzero A and B terms at the input scale. We
assume At,b = m3/2, A? = 2m3/2, A? = 0, BH = 2m3/2
and B = 0 at the input renormalization scale, Min. The
(m1/2, m1) plane for this case with Min = MGUT is shown in
the left panel of Fig. 2. This is the limiting case in which the
superGUT scenario reduces to an NUHM1 plane [37, 38, 99?
101] with m0 = 0 and A0 = m1. Note that the values of ?
and ? are irrelevant when taking Min = MGUT as there
is no running above the GUT scale in this case. There is
narrow band where the LSP is the lightest neutralino and
the electroweak symmetrybreaking conditions can be
satisfied, through which runs a blue relic density strip.12 At
low values of m1/2, the relic density is determined by stau
coannihilation [102?109], and the blue relic density strip lies
close to the boundary of the stau LSP region (shaded red).
At higher m1/2, the strip moves closer to the region with no
electroweak symmetry breaking (shaded pink) and becomes
a focuspoint strip [110?115]. The Higgs mass (shown by the
red dotdashed contours between the two excluded regions)
has acceptable values along much of the relic density strip.
On the other hand, the proton lifetime is too short as the
entire strip shown lies at or below the contour corresponding
to ? p = 0.001?1035 years (which appears as the black curve
that enters the allowed region at about 5 TeV at an angle to
the relic density strip). The right panel of Fig. 2 shows the
corresponding plane with the following choices of modular
weights: ?t,b = 1, ?? = 2, ?? = 0, ?H = 2 and ? = 0,
which correspond to A0 = B0 = 0. This exhibits many
features similar to the left panel. In particular, the relic density
and proton lifetime constraints are incompatible, motivating
our exploration of superGUT scenarios.
In Fig. 3, the model with all weights set to zero is assumed
again, but now with Min = 1016.5 GeV. The most dramatic
12 The relic density strip has been enhanced here and in subsequent
figures for better visibility by showing regions with ? h2 lies between
0.06 and 0.2.
1M2in = M24 GUT
2
1
112242
3
0.05
0
Fig. 2 Examples of (m1/2, m1) planes for Min = MGUT and tan ? = 7
when both Higgs 5plets are twisted. In the left panel, all the modular
weights ?i = ?i = 0, corresponding to At,b = m1, A? = 2m3/2,
A? = 0, BH = 2m1, and B = 0. In the right panel, the modular
weights are chosen to be ?t,b = 1, ?? = 2, ?? = 0, ?H = 2 and
? = 0, corresponding to A0 = B0 = 0. The shadings and contour
difference between this model and the previous GUT model
shown in the left panel of Fig. 2 is the disappearance of the
stau LSP region as Min is increased above the GUT scale,
an effect that was discussed in [57, 116?118]. In the
superGUT case even a small amount of running with m0 = 0
between MGUT and Min is sufficient to restore a neutralino
LSP. In the left panel of this figure, we have taken the Higgs
coupling, ? = 0.6, whereas in the right panel ? = 1, fixing
? = 10?5 in both panels. In this case, the relic density
strip (which is little changed from the GUT model) lies close
to the boundary where electroweak symmetry breaking is
not possible (shaded pink), and is similar to the focuspoint
region of the CMSSM [110?115].
Another very obvious difference between the left panel
of Figs. 2 and 3 is the value of the proton lifetime. With
Min = MGUT, the entire strip shown has a lifetime ? p <
1033 years, as it lies to the left of the contour labeled 0.01.
However, the proton lifetime is significantly longer in both
panels of Fig. 3, and there are acceptable parts of the relic
density strip where ? p > 0.066 ? 1035 years. Comparing the
two panels allows one to see the effect of increasing ? on
the proton lifetime. For ? = 0.6, the lifetime is sufficiently
long for m1/2 5 TeV, whereas for ? = 1 this is relaxed to
m1/2 2.5 TeV. Increasing ? much further is not possible
due to its effect on the Yukawa couplings, as discussed in [58].
In both cases, the Higgs masses are reasonably consistent
with 125 GeV, though due to the increased ?bending? of
0.001122
2
4
2
1
3
2
123 1
5
2
1
colors are the same as in Fig. 1. The width of the relic density region
(shaded blue) has been enhanced for better visibility. The pink shaded
region corresponds to parameter choices where the electroweak
vacuum conditions cannot be satisfied and radiative electroweak symmetry
breaking is not possible
the contours, the Higgs mass along the relic density strip is
slightly lower for the larger value of ?.13 The GM couplings
are also acceptably small: in the GUT case shown in the left
panel of Fig. 2 they are 1 across the plane, whereas in Fig. 3
c (m3/2/m1/2)2 is of order 0.05 all along the relic density
strip.
Since the strips with acceptable relic density in these
models resemble the familiar focuspoint region [110?115],
one can expect that the spinindependent elastic
scattering cross section on protons, ? SI, may be relatively large.
Concentrating on the right panel of Fig. 3, we have
computed ? SI at two points: (m1/2, m1) = (3100, 6000) GeV and
(4100, 8000) GeV. The resulting cross sections are ? SI =
(1.24 ? 0.77) ? 10?8 pb and (1.90 ? 1.19) ? 10?9 pb with
m? = 930 GeV and 1400 GeV, respectively, where we have
assumed ? N = 50 ? 8 MeV [119] and ?0 = 36 ? 7 MeV
[120]. The central value for the former point is slightly above
the recent LUX [121] and PandaX [122] bounds, but remains
acceptable when uncertainties in the computed cross sections
are taken into account. Furthermore, using nucleon matrix
elements computed with lattice simulations as in [123] would
reduce the predicted cross section by more than a factor 2
due to the smallness of strangequark content in a nucleon.
13 At higher Min, the bending of Higgs mass contours seen in Fig. 3 as
they approach the region with no radiative electroweak symmetry
breaking (shaded pink) becomes more severe, and the Higgs mass becomes
too low all along the relic density strip.
? = 0.6, ?' = 0.00001, tan ? = 7, ? > 0
M12i2n =1 1016.5 GeV123122 0.5
3
2
1
12
2
0.0120 6
5 3.06124
Fig. 3 Examples of (m1/2, m1) planes for Min = 1016.5 GeV when
both Higgs 5plets are twisted. All the modular weights ?i = ?i = 0,
corresponding to At,b = m1, A? = 2m3/2, A? = 0, BH = 2m1, and
B = 0. In both panels tan ? = 7 and ? = 10?5 with ? = 0.6 (left)
However, in both the cases studied one may anticipate a
positive signal in upcoming direct detection experiments such
as LUXZeplin and XENON1T/nT [124, 125].
We consider next the case with the modular weights ?t,b =
1, ?? = 2, ?? = 0, ?H = 2 and ? = 0, so that A0 = B0 =
0 for all A and B terms. The right panel of Fig. 2 shows
the (m1/2, m1) plane for Min = MGUT, which is similar to
that shown in the left panel when A and B terms are
nonzero. The A and B terms are seen to affect somewhat the
dependence on m1 of the Higgs mass and the position of
the relic density strip. The same case with A0 = B0 = 0
but Min = 1016.5 GeV is shown in the left panel of Fig. 4.
Comparing this with the right panel of Fig. 3, we see that the
proton lifetime shows little dependence on A0 and is similar
in the two cases shown. For larger Min = 1018 GeV with
A0 = B0 = 0, as shown in the right panel of Fig. 4, we see
that the relic density strip shifts to larger values of m1 and
the proton lifetime is somewhat longer. Much of the allowed
dark matter strip has an acceptably long proton lifetime. The
effect of adjusting the modular weights does not have a major
effect on the elastic scattering cross section.
We consider next the case where only one of the Higgs
5plets is twisted, so that
When H is twisted,
M122 in = 1120311122326.5 G0e.00.10V010.005
0.066
12
2
and ? = 1 (right). The shadings and contour colors are the same as
in Fig. 1. The width of the relic density region (shaded blue) has been
enhanced for better visibility. The pink shaded region corresponds to
parameter choices where the electroweak vacuum conditions cannot be
satisfied, and radiative electroweak symmetry breaking is not possible
whereas when H twisted,
Thus, in either case we have nonuniversal Aterms related
via the Yukawa couplings.
We consider first the case with twisted H . Examples of
(m1/2, m1) planes for Min = 1018 GeV are shown in Fig. 5.
In both panels, we have taken tan ? = 7, ? = 1, and ? =
10?5. Since H remains untwisted, we have m0 = m2 = 0
and, since the two Higgs soft masses are unequal, this is an
example of a superGUT NUHM2 model [37, 38, 101, 126,
127].14 The region where one obtains an acceptable relic
density could be expected from the upper left panel of Fig. 14
in [101], which shows an example of an (m1, m2) plane for
relatively low m1/2, m0 and tan ?. For m2 = 0, we expect
that there should be a funnel strip [11?15] where schannel
annihilation of the LSP through the heavy Higgs scalar and
pseudoscalar dominates the total cross section and m? ?
m A/2. This generally occurs when m21 < 0 at the input scale.
In the left panel of Fig. 5, we have taken ?t = 0, ?b = 1,
?? = 1, ?? = 0, ?H = 1 and ? = 0, so that all the A and B
terms vanish at the input scale. In the pink shaded region, the
electroweak symmetrybreaking (EWSB) conditions cannot
be satisfied as m2A < 0. Indeed, for m21 < 0, we see a blue
relic density strip above the shaded region. Whilst the proton
lifetime is sufficiently large for m1/2 1.8 TeV, the strip
14 The quoted sign of m1 actually represents the sign of m21 at the input
scale.
M122in = 1016.5 GeV
1213
22
0.5
0
1
2
5
m1/2 (GeV)
0.5
0
m1/2 (GeV)
0.6
6
0
5
2
1
4
2
1
?5
0.
?
1
?2000 ?
1.
5
m1/2 (GeV)
Fig. 4 Examples of (m1/2, m1) planes for Min = 1016.5 (left) and
1018 GeV (right) when both Higgs 5plets are twisted. In both cases the
modular weights are ?t,b = 1, ?? = 2, ?? = 0, ?H = 2 and ? = 0,
corresponding to A0 = B0 = 0, and we assume tan ? = 7, ? = 1 and
? = 10?5. The shadings and contour colors are the same as in Fig. 1.
The width of the relic density region (shaded blue) has been enhanced
for better visibility
Min = 1018 G0.01eV
Fig. 5 Examples of (m1/2, m1) planes for Min = 1018 GeV when only
H is twisted. In the left panel, the modular weights are ?t = 0, ?b = 1,
?? = 1, ?? = 0, ?H = 1 and ? = 0, so that all trilinear and bilinear
terms vanish. In the right panel, all the weights vanish, so that At = 0,
Ab = m1, A? = m1, A? = 0, BH = m1, and B = 0. In both panels
tan ? = 7, ? = 1 and ? = 10?5. The shadings and contour colors
are the same as in Fig. 1. The width of the relic density region (shaded
blue) has been enhanced for better visibility
extends (barely visibly) to mh = 123 GeV (shown by the red
dotdashed contours). In the right panel of this figure, we have
set all weights to zero, and therefore At = 0, Ab = m1, A? =
m1, A? = 0, BH = m1, and B = 0. Qualitatively, the two
figures are very similar. The strip extends to slightly larger
mh but, again, not much past 123 GeV. In both cases, At = 0
at the input scale and, although Ab = 0 in the right panel,
the dominant factor contributing to the Higgs mass is At . In
both panels c (m3/2/m1/2)2 ? ?0.25 in the allowed regions
of the parameter space. We see that the proton lifetime is
acceptably long when m1/2 1.7 TeV along the dark matter
strip. The elastic cross section near the end point of the relic
density strip where m1 ? ?3500 GeV is quite small: ? SI ?
1 ? 10?11 pb with m? ? 800 GeV, probably beyond the
Min = 1018 GeV
m1 = 3000 GeV
?1.5
?0.5
0.1
0
?1
1
2
3
0.
1
Fig. 6 Left panel the (At , m1/2) plane for tan ? = 7, ? = 1 and
? = 10?5 with Min = 1018 GeV, m0 = 0 and m1 = ?3000 GeV
when only H is twisted. Right panel the corresponding (m1/2, m1)
plane. Here, the modular weights are ?b = 1, ?? = 1, ?? = 0, ?H = 1
and ? = 0. For the left panel ?t varies and m1 is fixed while for the
right panel At = m1 (?t = ?1) and all other Ai = Bi = 0. The
shadings and contour colors are the same as in Fig. 1. The width of the relic
density region (shaded blue) has been enhanced for better visibility
reach of LUXZeplin and XENON1T/nT [124,125], though
still above the neutrino background level.
The Higgs mass can be increased slightly by turning on
the weight ?t controlling At . To determine the optimal value
for ?t , for all other Ai = 0 and Bi = 0, we scan over ?t .
In the left panel of Fig. 6 the resulting ( At , m1/2) plane for
fixed m1 = ?3000 GeV and m0 = m2 = 0 is shown. Once
again, the pink shaded region is excluded as m2A < 0 and
the constraints for electroweak symmetry breaking cannot be
satisfied. The blue line (enhanced here for visibility) shows
the position of the relic density funnel strip. We see that the
largest value of the Higgs mass obtained is slightly larger than
124 GeV, which is reached when At /m1 ? 1. The proton
lifetime is acceptably long for m1/2 1.8 TeV along the
dark matter strip, and the GM coupling shown by the green
lines is ?1.5 in this region. In the right panel, we show the
corresponding (m1/2, m1) plane with At = m1 and again all
other Ai = Bi = 0. Here we see that the funnel strip extends
to Higgs masses slightly larger than 124 GeV, where the
proton lifetime is about 1034 years. Points along the dark matter
strip with m1/2 1.7 TeV have an acceptably long proton
lifetime. In both cases, displayed, the elastic cross sections
are relatively small. Near the end point of the relic density
strip where m1 ? ?3000 GeV, we find ? SI ? 2 ? 10?11
pb with m? ? 950 GeV. Although this cross section is still
above the neutrino background, it may be difficult to detect
in the planned LUXZeplin and XENON1T/nT experiments.
Finally, we consider the effects of twisting H leaving H
untwisted. In this case, m0 = m1 = 0, and previous studies
lead us to expect the relic density strip to lie at positive values
of m2. Once again, we have taken tan ? = 7, ? = 1, and
2
? = 10?5. In the left panel of Fig. 7, we have taken ?t = 1,
?b = 0, ?? = 1, ?? = 0, ?H = 1 and ? = 0, so that all
A and B terms vanish at the input scale. In the pink shaded
region, the EWSB conditions cannot be satisfied, but in this
case it is because ?2 < 0. Just to the right of the excluded
region, we see the equivalent of the focuspoint strip, where
the LSP is mostly Higgsino. Still further to the right, we see
two closely spaced strips corresponding to the funnel region
with a mostly binolike LSP. For this choice of ? and ? , the
proton lifetime is sufficiently long if m1/2 1.8 TeV, but
the Higgs mass is 123 GeV unless m1/2 2.7 TeV. In the
right panel of Fig. 7, we again take all weights equal to 0, so
that At = m2, Ab = 0, A? = m2, A? = 0, BH = m2, and
B = 0. In this case, the pink shaded region has m2A < 0 and
we see the funnel strip running to values of mh > 125 GeV.
Comparing this with the left panel, we see the effect of the
nonzero value of At on mh . In both panels we see that points
along the dark matter strips with m1/2 1.7 TeV have an
acceptably long proton lifetime.
Since we have both a focuspoint strip and a funnel region,
there is more variation in the computed elastic cross
section. Corresponding to the left panel of Fig. 7, we considered
points at m2 = 4000 GeV with m1/2 2700 GeV (focus
point with m? 900 GeV) and m1/2 3200 GeV (funnel
with m? 1160 GeV). We found ? SI (2.2 ? 1.4) ? 10?8
pb and (1.2 ? 0.7) ? 10?10 pb respectively. At higher
m0 = 5000 GeV, the cross section on the focus point at
m1/2 1065 GeV drops to (6.4 ? 4.0) ? 10?9 pb and on
the funnel at m1/2 1530 GeV drops to (7.2 ? 4.5) ? 10?11
0.066
5
?0.
1230.
m1/2 (GeV)
Fig. 7 Examples of (m1/2, m2) planes for Min = 1018 when only H
is twisted. In the left panel, all trilinear and bilinear terms are zero. The
modular weights are ?t = 1, ?b = 0, ?? = 1, ?? = 0, ?H = 1 and
? = 0, corresponding to A0 = B0 = 0. In the right panel, all weights
are zero, so that At = m2, Ab = 0, A? = m2, A? = 0, BH = m2, and
B = 0. In both panels tan ? = 7, ? = 1 and ? = 10?5. The shadings
and contour colors are the same as in Fig. 1. The width of the relic
density region (shaded blue) has been enhanced for better visibility
pb. When the weights are set to zero as in the right panel
of Fig. 7, we have only a funnel strip and the cross section
is quite low. For (m1/2, m2) = (1920,3000), we find ? SI
(5.3 ? 3.3) ? 10?11 pb and for (m1/2, m2) = (2965,4400),
we find ? SI = (2.2 ? 1.4) ? 10?11 pb.
4 Discussion
Working within a noscale supergravity framework inspired
by string compactification scenarios, we have shown in this
paper that, if the matter and Higgs supermultiplets are all
untwisted, superGUT SU(5) models are unable to provide
simultaneously a long enough proton lifetime, a small enough
relic LSP density and an acceptable Higgs mass in the
framework of noscale supergravity, even in the presence of a
Giudice?Masiero term in the K?hler potential. However, all
of these phenomenological requirements can be reconciled
if one or both of the GUT Higgs fiveplets is twisted. We
have exhibited satisfactory solutions for various values of
the input superGUT scale Min, the GUT Yukawa couplings
that are important in the RGEs above the GUT scale, and the
modular weights of the various matter and Higgs fields. All
the examples shown assume tan ? = 7: significantly smaller
values of tan ? are largely excluded because mh is too small,
and significantly larger values of tan ? are largely excluded
because the proton lifetime is too short. Spinindependent
dark matter scattering may be observable in some of the cases
studied.
Although, as we have shown, many of the problems of the
minimal SU(5) GUT model may be resolved in the noscale
SU(5) superGUT, including rapid proton decay through
dimensionfive operators, in a manner compatible with the
dark matter density and the Higgs mass, other issues such as
neutrino masses/oscillations remain unresolved. Moreover,
the resolution of the minimal supersymmetric SU(5) GUT
problems within the superGUT and noscale supergravity
frameworks is quite constrained and somewhat contrived. It
also remains unclear how an SU(5) GUT model could be
embedded within string theory. However, our analysis may
point the way how this could be done successfully.
A natural alternative is the flipped SU(5) ? U(1)
framework proposed in [128?133], which resolves automatically
the problems mentioned above, and can be embedded with
string theory. Choosing even the simplest strict noscale
boundary conditions m0 = A0 = B0 = 0 at Min provides a
very interesting flipped SU(5) framework that satisfies all the
constraints from present lowenergy phenomenology,
including the relic dark matter density and the proton lifetime, and
makes interesting predictions for Run 2 of the LHC [134?
137]. Moreover, flipped SU(5) also contains a rationale for
Min > MGUT, since the final unification of the SU(5) and
U(1) gauge couplings could well occur at the string scale.
We therefore plan to consider the possibility of a noscale
flipped SU(5) superGUT in a forthcoming paper.
Acknowledgements The work of J. E. was supported in part by the UK
STFC via the research Grant ST/J002798/1. The work of D. V. N. was
supported in part by the DOE Grant DEFG0213ER42020 and in part
by the Alexander S. Onassis Public Benefit Foundation. The work of N.
N. and K. A. O. was supported in part by DOE Grant DESC0011842
at the University of Minnesota.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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