#  RUNHETC-2023-35
Old Ideas For New Physicists: II

Tom Banks
Department of Physics and NHETC
Rutgers University, Piscataway, NJ 08854
E-mail<EMAIL_ADDRESS>

###### Abstract

We review ideas relating the breaking of Supersymmetry in de Sitter (dS)
space-time to amplitudes in which gravitinos are reflected from the dS
horizon. A self consistent evaluation of such amplitudes leads to the estimate
$m_{3/2}=K\Lambda^{1/4}$, with a constant $K$ that is not easily calculable.
LHC bounds on superpartner masses, plus the structure of low energy effective
field theory then imply that there must be new superfields with standard model
quantum numbers in the region below $100$ TeV. Only one class of models, the
Pyramid Schemes, has been found, which is compatible with one loop coupling
unification. They have a new strongly coupled gauge sector with a confinement
scale in the multi TeV region. We outline the general predictions of these
models, which include Dirac gauginos and a novel solution of the strong CP
problem. The models all appear to have a ”little hierarchy problem”: the
expectation value of the standard model Higgs field, and its mass, must be a
factor of $10$ or more smaller than the masses of the second Higgs doublet or
standard model superpartners. Since all of these models contain uncertainties
from the strong coupling physics, as well as the physics of the dS horizon, it
is difficult to determine how serious this fine tuning problem is.

##  1 Introduction

All known consistent perturbative string compactifications to asymptotically
flat space have exact Supersymmetry (SUSY). With one class of exotic
exceptions[1], all well understood families of $AdS/CFT$ models with a zero
c.c. limit, are exactly SUSic. The real world, on the other hand, is not SUSic
and appears to have a very small positive cosmological constant (c.c.).
Neither perturbative string theory nor AdS/CFT applies to it directly. In this
note I will review some old ideas about the relation between the breaking of
SUSY in the real world and the value of the c.c.. In a nutshell, the claim is
that SUSic solutions of SUGRA generally have negative c.c. and zero c.c.
models are characterized by the existence of a discrete chiral R symmetry,
which sets the gravitino mass to zero. Note that in SUSic models in AdS space,
the gravitino is in a superconformal multiplet with the graviton and does not
have zero mass in effective field theory. The breaking of SUSY is intimately
connected with the breaking of this chiral R symmetry by reflection of
gravitinos from the cosmological horizon.

In[3][4] we explored the argument that de Sitter (dS) space was a system with
a finite dimensional Hilbert space, whose holographic degrees of freedom were
encoded in copies of the cutoff spinor bundle on the finite area holoscreen of
the cosmological horizon. Chiral symmetries are always broken by the boundary
conditions on space-times with boundaries. If we view an asymptotically flat
SUSic model as the limit of a model with positive c.c. then the breaking of R
symmetry should be associated with interactions in which the gravitino, or
other R charged particle, is reflected from the horizon. Any such amplitude
will have the form

$e^{-2m_{3/2}R}{\rm Tr}\rho_{h}VV^{\dagger},$ (1)

where $\rho_{h}$ is the horizon density matrix and $V$ the operator that
causes the horizon to absorb one gravitino. We’ve taken into account the
exponential falloff of two massive propagators in dS space, but not their
power law dependence. A gravitino of mass $m_{3/2}$ can remain within a Planck
length of the horizon for a proper time of order $m_{3/2}^{-1}$. During that
time, if it performs a random walk over the holographic screen, it can cover
an area of order $C(m_{3/2}M_{P})^{-1}$, if the length of a step is $L_{P}$.
Using the area law for entropy, the number of intermediate states that can
contribute to the operator product $VV^{\dagger}$ is
$e^{C(m_{3/2}/M_{P})^{-1}}$. This means that the value of the gravitino mass
is of order

$m_{3/2}\sim e^{-2m_{3/2}R}e^{C(m_{3/2}/M_{P})^{-1}}.$ (2)

For consistency, $m_{3/2}$ has to go to zero as $R$ goes to infinity. However,
if it goes to zero too rapidly, the second factor on the right hand side of
this estimate blows up. If it goes to zero so slowly that the first factor
dominates, then the estimate predicts that it vanishes exponentially, which is
again a contradiction. The only consistent behavior is for the two
exponentials to cancel, allowing corrections to the estimate to control the
falloff. This implies

$2m_{3/2}R=C(m_{3/2}/M_{P})^{-1}.$ (3)

Thus

$(m_{3/2}/M_{P})=B(RM_{P})^{-\frac{1}{2}},$ (4)

where

$B=(\frac{C}{2})^{\frac{1}{2}}.$ (5)

As we’ll note below, standard supergravity formulae then imply a SUSY breaking
scale below $100$ TeV. Thus we expect a renormalizable effective field theory
at the multiple TeV scale to describe SUSY breaking. It is impossible to
construct such a model, which is consistent with experiment, using just
standard model singlet fields and the SUSic standard model. Thus the
combination of the idea that SUSY breaking originates from gravitino
reflection off the cosmological horizon, with effective field theory, requires
us to add a new sector to effective field theory at energies close to the TeV
scale. The new sector must contain fields charged under the standard model
gauge group, and making it compatible with gauge coupling unification puts
major restrictions on what it can be. So far only one class of models has been
found that satisfies all the constraints. It is called the Pyramid Scheme,
because of the shape of its quiver diagram.

Our novel perspective on SUSY breaking also turns out to have interesting
implications for the strong CP problem. It turns out that in the limit in
which the c.c. vanishes, the model has an exact symmetry, which guarantees
that the only CP violating angle in the low energy effective theory is the CKM
angle. This symmetry is broken by the same gravitino diagrams that generate
the gravitino mass, but one can give plausible arguments that the CP symmetry
breaking terms are too small to generate a measureable strong CP angle.

##  2 A New Strongly Interacting Sector Close to the TeV Scale

Let us begin by recalling the estimate for the scale of Cosmological SUSY
breaking. In the limit of vanishing c.c. the gravitino mass is protected by a
discrete $Z_{N}$ R symmetry. The gravitino mass is generated by diagrams in
which the gravitino reflects off the horizon. These have the form

$e^{-2m_{3/2}R}\sum\langle s|T|n\rangle\langle
n|T^{\dagger}|s^{\prime}\rangle.$ (6)

Here we’ve assumed that the horizon is initially in a pure state . In a
Feynman diagram calculation the gravitino line strikes the horizon at some
point and then performs a random walk over the horizon. Since, in a self
consistent calculation, it is massive, the proper time of that walk is of
order $m_{3/2}^{-1}$ and so it covers an area of order $m_{3/2}^{-1}L_{P}$ .
Here we’ve assumed that the length of a step in the walk is the Planck length.
The Gibbons-Hawking area law tells us that the number of states contributing
positively in the sum is of order $e^{C(L_{P}m_{3/2})^{-1}}$. If we now
imagine that $m_{3/2}$ vanishes like some power of $R$ as
$R/L_{P}\rightarrow\infty$ then the only consistent scaling is

$m_{3/2}^{2}=(C/2)(RL_{P})^{-1}=(C/2)10^{-23}({\rm
GeV})^{2}=[\sqrt{C/2}10^{-2.5}{\rm eV}]^{2}.$ (7)

The standard supergravity (SUGRA) formula for the gravitino mass is

$m_{3/2}(1{\rm eV})^{-1}=\sqrt{F}(65{\rm TeV})^{-1}.$ (8)

So our formula predicts

$\sqrt{F}=.65{\rm TeV}\sqrt{\frac{C}{20}}.$ (9)

So one needs a fairly large value of $C$ to make this idea compatible with
experiment. Our understanding of the physics of the dS horizon, and of the way
in which effective field theory emerges as an approximation to the correct
theory of quantum gravity in dS space, is so primitive that one can easily
imagine that factors of $100$ could be missing from our estimates. In addition
we know that there is at least one other large energy scale, the scale of
coupling unification, evident in known physics. Neutrino masses, and many of
the models invoked to explain the hierarchies of quark and lepton Yukawa
couplings, suggest that the region $10^{14}-10^{16}$ GeV may have quite a bit
of structure in it. All of this could modify our crude estimate of the
gravitino mass.

If the SUSY breaking scale is this low, and there are no fields transforming
under the standard model gauge group in effective field theory below 100 TeV,
besides those of the minimal Supersymmetric Standard Model (MSSM), then it is
impossible to get a superpartner spectrum consistent with experiment. The only
renormalizable coupling of singlets to the MSSM is some linear combination of
singlet fields coupling to $H_{u}H_{d}$ and this can generate squark and
gluino masses only through loops, which will be much too small.

Once we add new low energy fields with standard model couplings, and a new
strong interaction scale to justify their existence, we have to worry about
coupling unification. So far only two general classes of models which seemed
to be consistent with coupling unification and the idea of low scale SUSY
breaking have been found: the Pentagon models[5] and the Pyramid schemes[6].
Detailed analysis[7] shows that the Pentagon models don’t work, while the
Pyramid schemes may be viable.

The quiver diagram of the Pyramid Schemes 1 is a three sided pyramid whose
base is the trinification model of Georgi and Glashow[8].

Figure 1: The quiver diagram of the Pyramid Schemes. Trianon fields connect
the apex of the Pyramid to the Trinification model of Glashow and Georgi at
the Base. Different versions of the Scheme have different Singlet couplings to
this quiver.

This unified model has gauge group $SU(3)_{3}\times SU(3)_{2}\times
SU(3)_{1}\ltimes Z_{3}$, where the $Z_{3}$ cyclicly permutes the three $SU(3)$
groups. Each field connecting the apex of the pyramid to the base is also a
fundamental + anti fundamental of the group at the apex of the pyramid, which
is also $SU(3)$. We call it $SU_{P}(3)$.

Color is the $SU_{3}(3)$ group of the base of the pyramid, and the $SU(2)$ of
the electroweak interactions is a Cartesian $SU(2)$ in $SU_{2}(3)$. Weak
hypercharge is a linear combination of the hypercharge generators of
$SU_{2}(3)$ and $SU_{1}(3)$. The 15 Weyl fermions of the standard model are
part of the $[27]$ dimensional representation of the base of the pyramid and
singlets under the apex. The other fields in the $[27]$ have masses of order
$10^{16}$ GeV and their absence at low energy gives rise to the running of the
couplings. Some of these fields might actually be the Higgs fields of the
MSSM, but in the effective Lagrangian below the unification scale, to one loop
order, we do not have to commit ourselves to such an identification.

The fields connecting the apex and the base are called trianons and form
complete multiplets of the trinification group at the base. As long as they
are degenerate, they do not effect the one loop running of standard model
couplings. The apex $SU_{P}(3)$ thus has $9$ flavors of $[3]+[\bar{3}]$ chiral
superfields. We can organize these fields into six $3\times 3$ matrices
$T_{i}$ and $\bar{T_{i}}$ where one index is the apex gauge quantum number and
the other labels the gauge quantum number under the $i$-th $SU(3)$ of the base
of the pyramid. The unique gauge invariant superpotential of dimension $\leq
4$ made from these fields is

$\sum_{i}[u_{i}{\rm det}(T_{i})+\bar{u}_{i}{\rm det}(\bar{T}_{i})].$ (10)

In addition to the supersymmetric standard model, we will introduce $P$
singlet fields $S_{p}$ whose role is to convey SUSY breaking to the rest of
the system. In the limit of vanishing c.c., the system has a discrete symmetry
group, one factor of which acts on the supercharges and guarantees that the
c.c. is zero, a fact which is otherwise unnatural in SUGRA even when SUSY is
preserved. We have a superpotential

$W_{\Lambda=0}=\sum_{pj}y_{pj}S_{p}{\rm tr}T_{j}\bar{T}_{j}+\sum_{i}[u_{i}{\rm
det}(T_{i})+\bar{u}_{i}{\rm
det}(\bar{T}_{i})+\beta_{p}S_{p}H_{u}H_{d}]+Y^{u}_{IJ}H_{u}\bar{U}_{I}Q_{J}+Y^{d}_{IJ}H_{d}\bar{D}_{I}Q_{J}+Y^{l}_{I}\bar{E}_{I}H_{d}L_{J}.$
(11)

The discrete symmetry group is chosen so that no other terms of dimension $4$
appear in the superpotential and that the only terms of dimension $5$ that
appear in the effective action, and which violate baryon or lepton number, are
those that give rise to neutrino masses

$W_{\nu}=\frac{G_{IJ}}{M}(L_{I}H_{u})(L_{J}H_{u}).$ (12)

All higher dimension operators are presumed to be scaled by the unification
scale $10^{16}$ GeV. $M$ should be slightly smaller and the explanation of
that is left for unification scale physics, as is the structure of the Yukawa
couplings.

In[9] it was shown that this can all be accomplished by a $Z_{8}$ R symmetry
with R charge assignments

$S_{i}=T_{i}=\bar{T_{i}}=6,\ \ \ \ Q_{I}=5,\ \ \ \ L_{I}=1,\ \ \ \ H_{d}=0,$
(13)

along with

$H_{u}=-4,\ \ \ \ \bar{U}_{I}=1,\ \ \ \ \bar{D}_{I}=-3,\ \ \ \ \bar{E}_{I}=1.$
(14)

In addition, we must invoke an ordinary discrete symmetry to forbid cubic
terms in the singlet fields.

###  2.1 Breaking the Discrete Symmetries

We now want to add explicit breaking of the discrete symmetries to mimic the
effects of the dS horizon. These should enforce the relation

$m_{3/2}=K\Lambda^{1/4},$ (15)

which we derived by considering self consistent gravitino mass generation via
reflection off the horizon. There is no reason for them to obey the genericity
constraints of traditional low energy effective field theory. We choose

$\delta W=W_{0}+\sum[m_{i}{\rm tr}\ (T_{i}\bar{T}_{i})+\mu_{p}^{2}S_{p}]+\mu
H_{u}H_{d}.$ (16)

If we take $\mu_{p}^{2}\sim\Lambda^{1/4}M_{P}$,
$m_{i}\sim\Lambda^{1/8}M_{P}^{1/2}$, and $W_{0}\sim\Lambda^{1/4}M_{P}^{2}$ ,
then we can satisfy the relationship between the gravitino mass and the c.c. .
Note that since these terms are non-generic, we avoid the theorem of Nelson
and Seiberg[10] about the necessity of R symmetry for spontaneous SUSY
breaking. The $F$ terms of the $S_{p}$ fields are non-vanishing at the
potential minimum where the VEVs of all fields vanish. Although this is not
the global minimum of the potential, the fact that SUSY is restored in the
limit that $\Lambda/m_{P}^{4}\rightarrow 0$ implies that the Coleman De Lucia
tunneling probability to a negative energy Big Crunch state will vanish
exponentially with the dS entropy. This is consistent[11][12] with an
interpretation of this transition as a temporary sojourn into a low entropy
state, as suggested by the Covariant Entropy Principle from a comparison of
the maximal causal diamonds in the two space-times.

In treating $\delta W$ as part of an effective Lagrangian below the
unification scale we must be aware of its origin in terms of diagrams with
gravitinos propagating out to the horizon. The local operators appearing in it
are therefore ”soft” and UV corrections to them much smaller than one would
estimate by power counting. This is why we don’t have to worry about ”large UV
renormalizations of the c.c.”. The phrase in scare quotes makes no sense in
terms of the underlying hypothesis that the c.c. is determined by the CEP, and
it is avoided in EFT calculations by remembering the softness of $\delta W$.
This is somewhat analogous to what happens in technicolor models, where a mass
term for a pseudo-goldstone boson is protected from large renormalization
because the boson is composite.

Another possible feature of $\delta W$ that is traceable to its unusual origin
is a novel solution to the strong CP problem. For $\Lambda=0$ the Pyramid
schemes incorporate the Peccei-Quinn solution to the strong CP problem. We
have written the most general Lagrangian, up to terms of dimension $4$ and $B$
conserving terms up to dimension $5$, consistent with a set of discrete
symmetries. It is easy to verify that all CP violating phases besides the QED
theta angle and the CKM phase can be rotated away. Let us assume that those
phases are generated by physics somewhat below the unification scale. An
example might be the scale of $10^{14.5}$ GeV that appears to be associated
with the dimension $5$ operator that gives rise to neutrino masses. The term
$\delta W$ is generated by interactions in which a gravitino is absorbed and
re-emitted by the degrees of freedom on the cosmological horizon. The
reflected gravitino trajectory is highly accelerated and so experiences an
Unruh temperature. If that temperature is higher than the energy scale
associated with CP violation, then the CP phases will average out and the
parameters in $\delta W$ will be real. The $\Lambda=0$ Lagrangian has a
Peccei-Quinn axion associated with an accidental $U(1)$ symmetry that is
broken by $\delta W$, but if the Unruh temperature is higher than the CP
violation scale, then $\delta W$ gives this axion a large mass without
introducing new CP violating phases into low energy physics. There is no
”axion quality problem” in this variation of the PQ mechanism, because terms
at the unifications scale, which might break the PQ symmetry are much smaller
than those coming from $\delta W$.

###  2.2 Approach to Phenomenology and Cosmology

The original analysis of these models[13][9] was done at a time before the
Higgs boson was discovered, and during which a variety of intriguing
indications for dark matter signals had been announced. Most of those have
gone away. In addition, Fischler and the author have discovered a holographic
model of inflation and dark matter[14] in which the dark matter particles are
primordial black holes (PBHs) which carry the lowest value of a discrete
$Z_{N}$ gauge symmetry charge. They have mass of order $M_{P}=10^{19}$ GeV.
This removes the necessity for the contortions of[9] which were necessary in
order to make perturbative gauge coupling running compatible with the
existence of a particle dark matter candidate. The models of[14] give a
plausible account of all features of the cosmology of the very early universe,
and depend on the nature of low energy particle physics only through the
number of species with masses below about $10^{8}$ GeV, and their spectrum.
Our analysis of the Pyramid schemes will concentrate only on particle physics.
The paper in earlier literature which we follow most closely is[15]. We differ
from that analysis mostly by insisting that all three pyrmabaryon number
violating Yukawa couplings appear in the Lagrangian for $\Lambda=0$. This
removes the bounds on the confinement scale $\Lambda_{3}$ from[9] as we recall
below.

We begin with a summary. $SU(3)$ SUSY gauge theory with $9$ flavors and the
indicated Yukawa couplings (we ignore the couplings to the singlets for the
moment), has an attractive line of fixed points at

$g_{P}^{2}=\frac{3}{4}u^{2},$ (17)

where all three Yukawas are equal and $g_{P}$ is the gauge coupling. This is a
one loop result. We assume that just below the unification scale the couplings
are in the domain of attraction of this fixed line and flow to it at a point
where the couplings are fairly strong. It’s not clear if we need to stay
within the domain of validity of the one loop calculation. We do so only in
order to have control of the calculation.

Since the trianons are in a full multiplet of the trinification group, the one
loop standard model coupling ratios flow in a way that is unchanged, if we
assume that the values just below the unification scale are perturbative, but
the QCD coupling increases more slowly in the IR, because we have, in effect,
3 more quark flavors. Thus, in order to match low energy physics we must start
from a larger gauge coupling at the unification scale. It is still consistent
to assume that the one loop approximation is adequate. This pattern of RG
flows continues until we hit the trianon mass thresholds. The trianon masses
all come from interactions with the horizon and we assume they’re roughly
equal.

##  3 RG Flow and The Effective Theory Below $\Lambda_{3}$

We begin with some unification scale values of all the gauge couplings and the
Yukawa couplings in $W_{\Lambda=0}$. These are assumed to be such that one
loop perturbation theory is a good approximation to RG running. For the
standard model couplings this means that the ratios run as they do in the
MSSM, while the individual couplings are less asymptotically free because of
the presence of the trianons. $g_{P}$ and the Yukawa couplings of the trianons
run rapidly to some point on the fixed line.

At some point we reach the scale of the trianon masses $m_{i}$ and $g_{P}$
begins to run towards strong coupling. $SU_{P}(3)$ confines at a scale
$\Lambda_{3}$ below $m_{i}$. Generically, the point on the fixed line to which
$g_{P}$ has run between the unification scale and the much lower scale $m_{i}$
is not weakly coupled. Therefore, $\Lambda_{3}\sim m_{i}$. $SU_{P}(3)$ SUSY
gauge theory with $9$ massive trianons may or may not have a meta-stable
Poincare invariant state, which preserves the standard model gauge group and
breaks SUSY, but we can certainly tune the couplings of the singlet fields
$S_{i}$ to guarantee the existence of such a state, and then tune $W_{0}$ so
that this state has a small positive c.c. in SUGRA, coinciding with the
observed value.

It should be emphasized again that these tunings have a different character in
models motivated by cosmological SUSY breaking than they would in conventional
low energy effective field theory. Conventionally, we are interested in the
lowest minimum of the potential energy density. In CSB we are modelling the
properties of localized excitations of a dS model of quantum gravity. The
$\Lambda=0$ limit of that model has excitations whose low energy scattering
matrix (for small enough numbers of particles) can be computed in some
particular $N=1$ SUGRA model with a discrete R symmetry, which guarantees that
the c.c. is zero in the low energy action. It might be that the low energy
action has a moduli space of SUSic, R symmetric solutions, but only one of
them has anything to do with the $\Lambda\neq 0$ model from which we began. In
that model, interactions with the horizon generate a non-generic
superpotential on the moduli space, including a constant $W_{0}$, which is
tuned to make the c.c. small and positive at one particular point in moduli
space. This is the point which is supposed to correspond to the model of the
real world. Now consider the effective potential on moduli space and suppose
it has a minimum lower than the one chosen to model the real world. Then
Coleman De Luccia instantons tunneling from the real world minimum to the
”true minimum” are above the Great Divide[12][11]. They correspond to
temporary sojourns of the typical state of the finite entropy dS vacuum state
into a highly improbable low entropy state. The return probability is governed
by the principle of detailed balance and is close to one. Thus, in treating
CSB with the tools of low energy field theory, we must choose minima of the
effective potential whose symmetry properties correspond to those we observe,
rather than search for a model where those properties emerge at the global
minimum of the potential.

For similar reasons we do not require the R breaking terms in the
superpotential to be generic, as an effective field theorist would. They are
there to ”retrofit” the EFT to an underlying model of quantum gravity whose
structure is quite different from QFT. For recent progress in understanding
the relation, see[4]. We’re still far from the point where we can compute the
terms in the EFT from an underlying gravitational model. We’ve learned from
perturbative string theory that we can trust EFT symmetry arguments when
$\Lambda=0$, but we also know that those arguments are wildly misleading at
the non-perturbative level. Almost all of the perturbative Calabi-Yau
compactifications of e.g. heterotic string theory with $N=1$ SUSY, which are
protected to all orders in string perturbation theory by non-renormalization
theorems based on holomorphy and axion shift symmetry, do not really exist as
non-perturbative models of quantum gravity. Our philosophy in this paper will
be to use conventional symmetry and genericity arguments for the $\Lambda=0$
terms in the EFT, but to choose the terms that arise from ”interactions with
the horizon” to fit phenomenological considerations, without paying attention
to EFT naturalness. It’s obvious that one must do this for the term $W_{0}$,
which fixes the c.c.. In the introduction we’ve shown how the model of the
origin of these terms from gravitino exchange might solve the strong CP
problem.

###  3.1 SUSY and Electroweak Symmetry Breaking

Just above the scale $\Lambda_{3}$, our model is weakly coupled. If we assume
that the squarks and sleptons don’t get tachyonic mass terms via their
perturbative couplings to the $SU_{P}(3)$ gauge theory, then the low energy
effective potential is

$V=g_{1}^{2}D_{1}^{2}+g_{2}^{2}D_{2}^{2}+(|\beta_{p}S_{p}+\mu|^{2})(|H_{u}|^{2}+|H_{d}|^{2})+K^{p\bar{q}}(S/\Lambda_{3},\bar{S}/\Lambda_{3})(\mu_{p}^{2}+\beta_{p}H_{u}H_{d}+\gamma_{pr}S_{r})(\mu_{p}^{*2}+\beta_{p}^{*}H_{u}^{*}H_{d}^{*}+\gamma_{pr}^{*}S_{r}^{*}).$
(18)

Recall that $\beta_{p}$ comes from the limiting model with $\Lambda=0$ , while
$\mu_{p}^{2}$ and $\gamma_{pr}$ come from interactions with the horizon and
vanish as $\Lambda/M_{P}^{4}\rightarrow 0$. The first two terms in the
potential are the electroweak D terms. They favor $\tan\beta=1$. Below
$\Lambda_{3}$ we cannot compute the Kahler metric of the singlet fields, even
if the couplings $\alpha_{pi}$ are small (as they should be if the model is to
remain perturbative up to the unification scale), since it depends on
correlation functions in the strongly coupled $SU_{P}(3)$ gauge theory.
However it is clear that, if we choose $\gamma_{pq}$ to have two zero
eigenvalues, SUSY is broken. This implies that the number of singlet fields
$P\geq 3$.

Before turning to a detailed discussion of constraints on the spectrum, let us
first outline the amount of variation that we can expect in the various
parameters, which would be determined by a more microscopic theory. The Kahler
metric, perturbatively in the parameters $y_{pi}$ is determined by

$\sum_{i}y_{pi}y_{qi}^{*}\langle M_{i}M_{i}^{*}\rangle,$ (19)

where $M_{i}$ is the singlet meson field constructed from the $i$-th trianon.
The parameters $y_{pi}$ come from some consistent model of quantum gravity in
asymptotically flat space with $N=1$ SUSY. Perturbative string theory gives us
continuous families of candidates, and F theory and M theory provide others.
Some of those have discrete R symmetries that guarantee that they will survive
non-perturbative corrections111Most don’t, despite being consistent to all
orders in string perturbation theory because of axion shift symmetries.. There
is no known catalog of these and there might be an infinite family of such
consistent models. One would then have to sift through those to find models
with the right low energy gauge group structure to give a Pyramid scheme
broken to the standard model times $SU_{P}(3)$, and determine how many low
energy singlets they contained and the values of the couplings $y_{pi}$ and
$\beta_{p}$. While current string theory technology is not yet capable of
performing this task, it is at least the type of task that traditional
perturbative string theorists worked on for decades.

The determination of the parameters $\mu_{p}^{2}$ and $\gamma_{pq}$ from a
more fundamental theory is more mysterious, and involves the poorly understood
theory of de Sitter space. According to the proposal of[4][18] this is a
finite quantum system, which converges to the ”string theory” S matrix in the
limit of vanishing c.c.. The general mathematical theory of limits suggests
that there will be many such systems, so that even given a choice of ”target”
SUSY S matrix, the values of the SUSY violating parameters have some allowed
range and we are far from understanding what it is.

All of the natural mass scales in our model, come from the multi TeV scale
determined by CSB. Even the confinement scale $\Lambda_{3}$ is close to this
scale because we’ve assumed that the fixed line value to which the $SU_{P}(3)$
coupling is driven is strong, so that confinement sets in at the trianon mass
scale. Nonetheless, we are going to assume that the Higgs VEVs are somewhat
smaller, so that $|H_{u}|^{2}+|H_{d}|^{2}$ takes on the value $(250{\rm
GeV})^{2}$ in the vacuum. This is the form of the ”little hierarchy problem”
in our model. We will comment below on the amount of fine tuning this
requires. In the approximation that the Higgs VEVs are zero, the singlet field
VEVS are determined by minimizing the effective potential

$V_{0}=K^{pq}(S,S*)(\mu_{p}^{2}+\delta_{p}^{3}\gamma S_{3})(\mu_{q}^{*\
2}+\delta_{p}^{3}\gamma^{*}S^{*\ 3}).$ (20)

SUSY is broken, the positive vacuum energy is canceled by the constant $W_{0}$
in the superpotential to give the observed valued of the c.c., and the singlet
fields all have multi TeV scale VEVs. The dimensionless Kahler metric has some
fixed numerical value at the vacuum values of the scalars.

Next we examine the quadratic terms in the Higgs fields, expanded around these
singlet VEVS, $S_{P}^{0}$. These have the form

${\cal
M}^{2}H_{u}H_{d}+h.c.+|\beta_{p}S_{p}^{0}+\mu|^{2}(|H_{u}|^{2}+|H_{d}|^{2}),$
(21)

where ${\cal M}$ is of order the multi-TeV scale . The dynamics of the
$SU_{P}(3)$ gauge theory preserves CP and we’ve argued that the horizon
physics that determines the dimensionful parameters in $\delta W$ is also CP
conserving, so ${\cal M}^{2}$ should be real. It is easy to see that the
potential is minimized when the phase of $H_{u}H_{d}$ is zero and the VEVs are
oriented so the the $U(1)$ of electromagnetism is unbroken. We are left with
quadratic terms

${\cal M}^{2}H_{u}H_{d}+|\tilde{\mu}|^{2}(H_{u}^{2}+H_{d}^{2}),$ (22)

where the values of the field are real. The first term is an anti-symmetric
matrix which has equal positive and negative eigenvalues. There is thus an
$SU(2)\times U(1)$ breaking second order phase transition as we vary the
parameters. Near this transition, on the symmetry breaking side, both the size
of the symmetry breaking VEV and the mass of the light Higgs field are much
smaller than the nominal multi-TeV scale of the action. Near the transition,
fluctuation corrections also become important and we know[19][20] that in fact
the transition is really a fluctuation induced first order phase transition.
The symmetry breaking VEV jumps to a finite value as the parameters are
varied. To get the actual values of the Z and Higgs boson mass right we have
to adjust two parameters, but only one of them has to be finely tuned, to the
percent level. The fields associated with the second Higgs multiplet all get
multi TeV scale masses, and the parameter $\tilde{\mu}$ is the mass that pairs
up the Weyl fermions in the two Higgs multiplets. We will discuss gauginos,
squarks and sleptons in the next subsection.

###  3.2 Standard Model Gaugino Masses

The confining $SU_{P}(3)$ gauge theory has composite chiral superfields
$M^{a}_{i}$ which transform in the adjoint representation of the standard
model gauge group. There are non-zero two point functions

$\langle M^{a}_{i}M^{b}_{i}\rangle,$ (23)

and

$\langle D_{\alpha}M^{a}_{i}W^{a}_{\beta i},\rangle$ (24)

mixing these fields with the standard model vector field strengths. The second
of these has the scaling properties of a hadron magnetic moment in QCD. It is
proportional to $g_{i}(\Lambda_{3})$ with all other dependence determined by
dimensional analysis with no extra powers of $\pi$ . Since $\Lambda_{3}$ and
the trianon masses are of the same order of magnitude, this mixing will
produce Dirac gaugino masses, once we plug in non-zero $F$ terms for the
singlets.

$m^{1/2}_{i}=C_{i}g_{i}(\Lambda_{3})\sqrt{F},$ (25)

where we expect the $C_{i}$ to be of order $1$. The idea is that, treating the
$y_{ip}$ couplings as numbers of order $1$, there will be terms in the
effective action of the form

$\int d^{4}\theta g_{i}f_{i}(S/\Lambda_{3})D_{\alpha}M^{i}W_{\alpha}^{i},$
(26)

for the $SU(i)$ subgroup of the standard model. We’ve seen that the F terms of
the singlet fields get expectation values of order $\Lambda_{3}^{2}$, so this
gives Dirac masses of the required order of magnitude. The Majorana masses of
the gauginos are much smaller, of order $\frac{\alpha_{i}F}{4\pi\Lambda_{3}}$,
with $F\sim\Lambda_{3}^{2}$ for reasons explained above. They arise from one
loop diagrams as explained in[16].

One loop sparticle masses coming from the Dirac mass are UV convergent, flavor
blind, and of order $\sqrt{\frac{\alpha_{i}}{4\pi}}m^{1/2}_{i}$. This is a
well known result from the literature on ”supersoft” SUSY breaking[17]. The
”UV divergent” contribution from the small Majorana mass is much smaller,
because the logarithmic divergence is cut off at the low value $\sqrt{F}$.
SUSY breaking in CSB is soft because it comes from diagrams with gravitinos
exchanged with the horizon. The most significant bounds on the scale
$\sqrt{F}$ thus come from the experimental bounds on gaugino, squark and
slepton masses, which are on the order of $1-3$ TeV. Therefore $\sqrt{F}$ in
the multi TeV range is compatible with experiments, modulo unknown strong
$SU_{P}(3)$ ”order one” factors.

##  4 Conclusions

Very general considerations in quantum gravity lead one to the conclusion that
SUSY is necessary for asymptotically flat space-time and that the cosmological
constant is a model parameter that must be set rather than calculated[4].
Observation seems to lead to the conclusion that our own universe will
asymptote to de Sitter space, with a cosmological horizon radius of about
$10^{61}$ in Planck units. SUGRA teaches us that vanishing c.c. requires not
just SUSY, but a discrete[2] R symmetry, if the model is to have a one
parameter set of dS perturbations. Those perturbations break SUSY and the R
symmetry in tandem, to keep the c.c. small. The cosmological horizon naturally
breaks the chiral R symmetry, giving the gravitino a mass222Note that in AdS
space the reflection of gravitinos from the boundary preserves both SUSY and
the R symmetry..

The strategy of CSB is to find an effective field theory implementation of
these quantum gravity rules, which is compatible with existing experiments,
and the apparent unification of standard model couplings at about $10^{16}$
GeV. So far, only a single class of models, the Pyramid Schemes, has been
found, which may accomplish this task. One begins with a spectrum compatible
with trinification, with extra vector-like trianons under each of the
$SU_{i}(3)$ of the trinification group, and also transforming like $3+\bar{3}$
under a new $SU_{P}(3)$. In the zero c.c. limit, this gauge theory flows to a
conformal fixed line at low energy, generically at a strongly coupled point.
At this stage of model building, traditional string theory techniques could be
extremely useful. Are there string constructions that lead to trinification
plus this extra structure? Can they have the right number of standard model
generations, Yukawa and neutrino textures, etc. to fit experiment? Do we have
a discrete R symmetry with the properties assumed in this paper? All of those
things should be relatively unaffected by CSB.

The other aspect of these models that could be attacked by traditional string
theory methods is the question of the origin of the singlets. Are they
remnants of trinification multiplets? Do they come from some other aspect of
string models? What determines their number and the values of their couplings
$y_{ip}$ to the trianons and $\beta_{p}$ to $H_{u}H_{d}$? These are all
$\Lambda=0$ questions, within the realm of conventional string/M-theory model
building.

The values of the trianon masses $m_{i}$, the $\mu$ and $\mu_{p}^{2}$
parameters, and the degenerate matrix $\gamma_{pq}$ will have to be extracted
from a much more quantitative theory of de Sitter space than anything we have
at present. The degeneracy of $\gamma$ is the most bizarre requirement from an
effective field theory point of view, but it follows from the SUSY violating
properties of dS space. It is interesting that once one has imposed that
condition, the observed standard model symmetry breaking pattern emerges
automatically, once we assume that $SU(2)\times U(1)$ is broken. The model
naturally allows for a weakly first order phase transition and we have to fine
tune a single parameter in order to obtain the correct value of the electoweak
gauge boson masses and the Higgs boson mass, while keeping the rest of the
spectrum consistent with LHC bounds. It is difficult to assess the complex of
issues that are known as the ”little hierarchy problem” because our parameters
come from the mysterious interactions with the horizon and also involve
strongly coupled $SU_{P}(3)$ physics.

There seem to be three roads to making future progress on these models. The
first is to delve back into traditional string phenomenology to find a model
that fits with the requirements of the vanishing $\Lambda$ limit of the
Pyramid schemes: the Pyramid gauge group broken at the proper unification
scale to $SU_{P}(3)\times G_{std}\times R$ where $R$ is the discrete group
that forbids all the couplings we’ve left out of the model. That model should
also explain the origin of the singlets and calculate the couplings $y_{ip}$,
$\beta_{p}$, and the unification scale values of the gauge and Yukawa
couplings of the Pyramid scheme. This model should also explain the origins of
the Yukawa couplings of quarks and leptons and of the dimension $5$ operators
that give rise to neutrino masses. All of this indicates a lot of structure in
the range $10^{14}-10^{16}$ GeV, which must be accounted for by a theory in
asymptotically flat space time and explains a lot of the phenomenology of
particle physics.

Next, we must find a quantitative solution of the strongly coupled $SU_{P}(3)$
SUSY gauge theory, with quark masses of order the confinement scale and Yukawa
couplings to $3$ or more chiral singlets. This is of course a major
undertaking, but it is the kind of task we’ve been contemplating since the
1970s.

The hardest of the three paths is to make progress on a more quantitative
theory of de Sitter space, in order to calculate the parameters
$\mu_{p}^{2},\mu,\gamma_{pq}$.

It is worth stressing again that the implication of our approach to SUSY
breaking is that the structure of the neutrino, quark and charged lepton mass
matrices, as well as the origin of CP violation, is assumed to be well
described by a supersymmetric theory with vanishing c.c., which is the origin
of a set of energy scales between $10^{14}-10^{16}$ GeV. The high scale is the
scale of coupling unification and the lower scales are associated with
neutrino masses. Powers of ratios between them account for the structure of
the CKM and neutrino mass matrices. In particular, CP violation exists, only
because of the presence of the lower scale. In this case, if the Unruh
temperature of gravitinos reflected from the dS horizon is $>10^{16}$ GeV, we
have a natural solution of the strong CP problem. This picture implies that
some of the tools of old fashioned string phenomenology might be useful in
describing the real world. They can determine the supersymmetric limit to
which our universe would tend if we could tune the c.c. to zero.

##  5 Acknowledgements

We thank Profs. H. Haber and particularly S. Thomas for useful conversations
about SUSY phenomenology and the proper treatment of the low energy Higgs
sector. This paper was completed while the author was attending the workshop
Gravity: New Perspectives from Strings and Higher Dimensions, at the Centro de
Ciencias Pedro Pascual, Benasque Spain. The author thanks, R. Emparan,V.
Hubeny, M. Rangamani, the organizers, and the staff of the center, for
providing a venue where a lot of exciting scientific work could get done. This
work was supported in part by the U.S. Dept. of Energy under Grant DE-
SC0010008.

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