Blueprints of the No-Scale Multiverse at the LHC
We present a contemporary perspective on the String Landscape and the Multiverse of plausible string, M- and F-theory vacua. In contrast to traditional statistical classifications and capitulation to the anthropic principle, we seek only to demonstrate the existence of a non-zero probability for a universe matching our own observed physics within the solution ensemble. We argue for the importance of No-Scale Supergravity as an essential common underpinning for the spontaneous emergence of a cosmologically flat universe from the quantum “nothingness”. Concretely, we continue to probe the phenomenology of a specific model which is testable at the LHC and Tevatron. Dubbed No-Scale -, it represents the intersection of the Flipped Grand Unified Theory (GUT) with extra TeV-Scale vector-like multiplets derived out of F-theory, and the dynamics of No-Scale Supergravity, which in turn imply a very restricted set of high energy boundary conditions. By secondarily minimizing the minimum of the scalar Higgs potential, we dynamically determine the ratio of up- to down-type Higgs vacuum expectation values (VEVs), the universal gaugino boundary mass GeV, and consequently also the total magnitude of the GUT-scale Higgs VEVs, while constraining the low energy Standard Model gauge couplings. In particular, this local minimum minimorum lies within the previously described “golden strip”, satisfying all current experimental constraints. We emphasize, however, that the overarching goal is not to establish why our own particular universe possesses any number of specific characteristics, but rather to tease out what generic principles might govern the superset of all possible universes.
pacs:11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv
The number of consistent, meta-stable vacua of string, M- or (predominantly) F-theory flux compactifications which exhibit broadly plausible phenomenology, including moduli stabilization and broken supersymmetry Bousso and Polchinski (2000); Giddings et al. (2002); Kachru et al. (2003); Susskind (2003); Denef and Douglas (2004, 2005), is popularly estimated Denef et al. (2004, 2007) to be of order . It is moreover currently in vogue to suggest that degeneracy of common features across these many “universes” might statistically isolate the physically realistic universe from the vast “landscape”, much as the entropy function coaxes the singular order of macroscopic thermodynamics from the chaotic duplicity of the entangled quantum microstate. We argue here though the counter point that we are not obliged a priori to live in the likeliest of all universes, but only in one which is possible. The existence merely of a non-zero probability for our existence is sufficient.
We indulge for this effort the fanciful imagination that the “Multiverse” of string vacua might exhibit some literal realization beyond our own physical sphere. A single electron may be said to wander all histories through interfering apertures, though its arrival is ultimately registered at a localized point on the target. The journey to that destination is steered by the full dynamics of the theory, although the isolated spontaneous solution reflects only faintly the richness of the solution ensemble. Whether the Multiverse be reverie or reality, the conceptual superset of our own physics which it embodies must certainly represent the interference of all navigable universal histories.
Surely many times afore has mankind’s notion of the heavens expanded - the Earth dispatched from its central pedestal in our solar system and the Sun rendered one among some hundred billion stars of the Milky Way, itself reduced to one among some hundred billion galaxies. Finally perhaps, we come to the completion of our Odyssey, by realizing that our Universe is one of at least so possible, thus rendering the anthropic view of our position in the Universe (environmental coincidences explained away by the availability of solar systems) functionally equivalent to the anthropic view of the origin of the Universe (coincidences in the form and content of physical laws explained away by the availability, through dynamical phase transitions, of universes). Nature’s bounty has anyway invariably trumped our wildest anticipations, and though frugal and equanimous in law, she has spared no extravagance or whimsy in its manifestation.
Our perspective should not be misconstrued, however, as complacent retreat into the tautology of the weak anthropic principle. It is indeed unassailable truism that an observed universe must afford and sustain the life of the observer, including requisite constraints, for example, on the cosmological constant Weinberg (1987) and gauge hierarchy. Our point of view, though, is sharply different; we should be able to resolve the cosmological constant and gauge hierarchy problems through investigation of the fundamental laws of our (or any single) Universe, its accidental and specific properties notwithstanding, without resorting to the existence of observers. In our view, the observer is the output of, not the raison d’être of, our Universe. Thus, our attention is advance from this base camp of our own physics, as unlikely an appointment as it may be, to the summit goal of the master theory and symmetries which govern all possible universes. In so seeking, our first halting forage must be that of a concrete string model which can describe Nature locally.
Ii The Ensemble Multiverse
The greatest mystery of Nature is the origin of the Universe itself. Modern cosmology is relatively clear regarding the occurrence of a hot big bang, and subsequent Planck, grand unification, cosmic inflation, lepto- and baryogenesis, and electroweak epochs, followed by nucleosynthesis, radiation decoupling, and large scale structure formation. In particular, cosmic inflation can address the flatness and monopole problems, explain homogeneity, and generate the fractional anisotropy of the cosmic background radiation by quantum fluctuation of the inflaton field Guth (1981); Linde (1982); Albrecht and Steinhardt (1982); Ellis et al. (1983a); Nanopoulos et al. (1983). A key question though, is from whence the energy of the Universe arose. Interestingly, the gravitational field in an inflationary scenario can supply the required positive mass-kinetic energy, since its potential energy becomes negative without bound, allowing that the total energy could be exactly zero.
Perhaps the most striking revelation of the post-WMAP Spergel et al. (2003, 2007); Komatsu et al. (2010) era is the decisive determination that our Universe is indeed globally flat, i.e. with the net energy contributions from baryonic matter , dark matter , and the cosmological constant (dark energy) finely balanced against the gravitational potential. Not long ago, it was possible to imagine the Universe, with all of its physics intact, hosting any arbitrary mass-energy density, such that “” would represent a super-critical cosmology of positive curvature, and “” the sub-critical case of negative curvature. In hindsight, this may come to seem as naïve as the notion of an empty infinite Cartesian space. The observed energy balance is highly suggestive of a fundamental symmetry which protects the “” critical solution, such that the physical constants of our Universe may not be divorced from its net content.
This null energy condition licenses the speculative connection ex nihilo of our present universe back to the primordial quantum fluctuation of an external system. Indeed, there is nothing which quantum mechanics abhors more than nothingness. This being the case, an extra universe here or there might rightly be considered no extra trouble at all! Specifically, it has been suggested Guth (1981); Linde (1982); Albrecht and Steinhardt (1982); Steinhardt (1984); Vilenkin (1983) that the fluctuations of a dynamically evolved expanding universe might spontaneously produce tunneling from a false vacuum into an adjacent (likely also false) meta-stable vacuum of lower energy, driving a local inflationary phase, much as a crystal of ice or a bubble of steam may nucleate and expand in a super-cooled or super-heated fluid during first order transition. In this “eternal inflation” scenario, such patches of space will volumetrically dominate by virtue of their exponential expansion, recursively generating an infinite fractal array of causally disconnected “Russian doll” universes, nesting each within another, and each featuring its own unique physical parameters and physical laws.
From just the specific location on the solution “target” where our own Universe landed, it may be impossible to directly reconstruct the full theory. Fundamentally, it may be impossible even in principle to specify why our particular Universe is precisely as it is. However, superstring theory and its generalizations may yet present to us a loftier prize - the theory of the ensemble Multiverse.
Iii The Invariance of Flatness
More important than any differences between various possible vacua are the properties which might be invariant, protected by basic symmetries of the underlying mechanics. We suppose that one such basic property must be cosmological flatness, so that the seedling universe may transition dynamically across the boundary of its own creation, maintaining a zero balance of some suitably defined energy function. In practice, this implies that gravity must be ubiquitous, its negative potential energy allowing for positive mass and kinetic energy. Within such a universe, quantum fluctuations may not again cause isolated material objects to spring into existence, as their net energy must necessarily be positive. For the example of a particle with mass on the surface of the Earth, the ratio of gravitational to mass energy is more than nine orders of magnitude too small
where is the gravitational constant, is the speed of light, and and are the mass and radius of the Earth, respectively. Even in the limiting case of a Schwarzschild black hole of mass , a particle of mass at the horizon has a gravitational potential which is only half of that required.
It is important to note that while the energy density for the gravitational field is surely negative in Newtonian mechanics, the global gravitational field energy is not well defined in general relativity. Unique prescriptions for a stress-energy-momentum pseudotensor can be formulated though, notably that of Landau and Lifshitz. Any such stress-energy can, however, be made to vanish locally by general coordinate transformation, and it is not even entirely clear that the pseudotensor so applied is an appropriate general relativistic object. Given though that Newtonian gravity is the classical limit of general relativity, it is reasonable to suspect that the properly defined field energy density will be likewise also negative, and that inflation is indeed consistent with a correctly generalized notion of constant, zero total energy.
A universe would then be in this sense closed, an island unto itself, from the moment of its inception from the quantum froth; only a universe in toto might so originate, emerging as a critically bound structure possessing profound density and minute proportion, each as accorded against intrinsically defined scales (the analogous Newton and Planck parameters and the propagation speed of massless fields), and expanding or inflating henceforth and eternally.
Iv The Invariance of No-Scale SUGRA
Inflation, driven by the scalar inflaton field is itself inherently a quantum field theoretic subject. However, there is tension between quantum mechanics and general relativity. Currently, superstring theory is the best candidate for quantum gravity. The five consistent ten dimensional superstring theories, namely heterotic , heterotic , Type I, Type IIA, Type IIB, can be unified by various duality transformations under an eleven-dimensional M-theory Witten (1995), and the twelve-dimensional F-theory can be considered as the strongly coupled formulation of the Type IIB string theory with a varying axion-dilaton field Vafa (1996). Self consistency of the string (or M-, F-) algebra implies a ten (or eleven, twelve) dimensional master spacetime, some elements of which – six (or seven, eight) to match our observed four large dimensions – may be compactified on a manifold (typically Calabi-Yau manifolds or manifolds) which conserves a requisite portion of supersymmetric charges.
The structure of the curvature within the extra dimensions dictates in no small measure the particular phenomenology of the unfolded dimensions, secreting away the “closet space” to encode the symmetries of all gauged interactions. The physical volume of the internal spatial manifold is directly related to the effective Planck scale and basic gauge coupling strengths in the external space. The compactification is in turn described by fundamental moduli fields which must be stabilized, i.e. given suitable vacuum expectation values (VEVs). The famous example of Kaluza and Klein prototypes the manner in which general covariance in five dimensions is transformed to gravity plus Maxwell theory in four dimensions when the transverse fifth dimension is cycled around a circle. The connection of geometry to particle physics is perhaps nowhere more intuitively clear than in the context of model building with -branes, where the gauge structure and family replication are related directly to the brane stacking and intersection multiplicities. The Yukawa couplings and Higgs structure are in like manners also specified, leading after radiative symmetry breaking of the chiral gauge sector to low energy masses for the chiral fermions and broken gauge generators, each massless in the symmetric limit.
From a top-down view, Supergravity (SUGRA) is an ubiquitous infrared limit of string theory, and forms the starting point of any two-dimensional world sheet or D-dimensional target space action. The mandatory localization of the Supersymmetry (SUSY) algebra, and thus the momentum-energy (space-time translation) operators, leads to general coordinate invariance of the action and an Einstein field theory limit. Any available flavor of Supergravity will not however suffice. In general, extraneous fine tuning is required to avoid a cosmological constant which scales like a dimensionally suitable power of the Planck mass. Neglecting even the question of whether such a universe might be permitted to appear spontaneously, it would then be doomed to curl upon itself and collapse within the order of the Planck time, for comparison about seconds in our Universe. Expansion and inflation appear to uniquely require properties which arise naturally only in the No-Scale SUGRA formulation Cremmer et al. (1983); Ellis et al. (1984a, b, c); Lahanas and Nanopoulos (1987).
SUSY is in this case broken while the vacuum energy density vanishes automatically at tree level due to a suitable choice of the Kähler potential, the function which specifies the metric on superspace. At the minimum of the null scalar potential, there are flat directions which leave the compactification moduli VEVs undetermined by the classical equations of motion. We thus receive without additional effort an answer to the deep question of how these moduli are stabilized; they have been transformed into dynamical variables which are to be determined by minimizing corrections to the scalar potential at loop order. In particular, the high energy gravitino mass , and also the proportionally equivalent universal gaugino mass , will be established in this way. Subsequently, all gauge mediated SUSY breaking soft-terms will be dynamically evolved down from this boundary under the renormalization group Giudice and Rattazzi (1999), establishing in large measure the low energy phenomenology, and solving also the Flavour Changing Neutral Current (FCNC) problem. Since the moduli are fixed at a false local minimum, phase transitions by quantum tunneling will naturally occur between discrete vacua.
We conjecture, for the reasons given prior, that the No-Scale SUGRA construction could pervade all universes in the String Landscape with reasonable flux vacua. This being the case, intelligent creatures elsewhere in the Multiverse, though separated from us by a bridge too far, might reasonably so concur after parallel examination of their own physics. Moreover, they might leverage via this insight a deeper knowledge of the underlying Multiverse-invariant master theory, of which our known string, M-, and F-theories may compose some coherently overlapping patch of the garment edge. Perhaps we yet share appreciation, across the cords which bind our 13.7 billion years to their corresponding blink of history, for the common timeless principles under which we are but two isolated condensations upon two particular vacuum solutions among the physical ensemble.
V An Archetype Model Universe
Though we engage in this work lofty and speculative questions of natural philosophy, we balance abstraction against the measured material underpinnings of concrete phenomenological models with direct and specific connection to tested and testable particle physics. If the suggestion is correct that eternal inflation and No-Scale SUGRA models with string origins together describe what is in fact our Multiverse, then we must as a prerequisite settle the issue of whether our own phenomenology can be produced out of such a construction.
In the context of Type II intersecting D-brane models, we have indeed found one realistic Pati-Salam model which might describe Nature as we observe it Cvetic et al. (2004); Chen et al. (2008a, b). If only the F-terms of three complex structure moduli are non-zero, we also automatically have vanishing vacuum energy, and obtain a generalized No-Scale SUGRA. It seems to us that the string derived Grand Unified Theories (GUTs), and particularly the Flipped models Barr (1982); Derendinger et al. (1984); Antoniadis et al. (1987), are also candidate realistic string models with promising predictions that can be tested at the Large Hadron Collider (LHC), the Tevatron, and other future experiments.
In the latter case, the Flipped gauge symmetry can be broken down to the SM gauge symmetry by giving VEVs to one pair of the Higgs fields and with quantum numbers and , respectively. The doublet-triplet splitting problem can be solved naturally via the missing partner mechanism Antoniadis et al. (1987). Historically, Flipped models have been constructed systematically in the free fermionic string constructions at Kac-Moody level one Antoniadis et al. (1987); Antoniadis et al. (1988a, b); Antoniadis et al. (1989); Lopez et al. (1993). To address the little hierarchy problem between the unification scale and the string scale, the Testable Flipped model class was proposed, which introduces extra TeV-scale vector-like particles Jiang et al. (2007). Models of this type have been constructed locally as examples of F-theory model building Beasley et al. (2009a, b); Donagi and Wijnholt (2008a, b); Jiang et al. (2009, 2010), and dubbed - Jiang et al. (2009, 2010) within that context.
Most recently, we have studied No-Scale extensions of the prior in detail Li et al. (2011a, b, 2010a), emphasizing the essential role of the tripodal foundation formed by the -lipped GUT Barr (1982); Derendinger et al. (1984); Antoniadis et al. (1987), two pairs of TeV scale vector-like multiplets with origins in -theory Jiang et al. (2007, 2009, 2010); Li et al. (2011c) model building, and the boundary conditions of No-Scale Supergravity Cremmer et al. (1983); Ellis et al. (1984a, b, c); Lahanas and Nanopoulos (1987). It appears that the No-Scale scenario, particularly vanishing of the Higgs bilinear soft term , comes into its own only when applied at an elevated scale, approaching the Planck mass. GeV, the point of the second stage unification, emerges in turn as a suitable candidate scale only when substantially decoupled from the primary GUT scale unification of via the modification to the renormalization group equations (RGEs) from the extra -theory vector multiplets.
In particular, we have systematically established the hyper-surface within the , top quark mass , gaugino mass , and vector-like particle mass parameter volume which is compatible with the application of the simplest No-Scale SUGRA boundary conditions Cremmer et al. (1983); Ellis et al. (1984a, b, c); Lahanas and Nanopoulos (1987), particularly the vanishing of the Higgs bilinear soft term at the ultimate - unification scale Li et al. (2011a, b). We have demonstrated that simultaneous adherence to all current experimental constraints, most importantly contributions to the muon anomalous magnetic moment Bennett et al. (2004), the branching ratio limit on Barberio et al. (2007); Misiak et al. (2007), and the 7-year WMAP relic density measurement Spergel et al. (2003, 2007); Komatsu et al. (2010), dramatically reduces the allowed solutions to a highly non-trivial “golden strip” with , , , and , effectively eliminating all extraneously tunable model parameters, where the consonance of the theoretically viable range with the experimentally established value :20 (2009) is an independently correlated “postdiction”. Finally, taking a fixed -boson mass, we have dynamically determined the universal gaugino mass and fixed via the “Super No-Scale” mechanism Li et al. (2010a), that being the secondary minimization, or minimum minimorum, of the minimum of the Higgs potential for the electroweak symmetry breaking (EWSB) vacuum.
These models are moreover quite interesting from a phenomenological point of view Jiang et al. (2009, 2010). The predicted vector-like particles can be observed at the Large Hadron Collider, and the partial lifetime for proton decay in the leading channels falls around years, testable at the future Hyper-Kamiokande Nakamura (2003) and Deep Underground Science and Engineering Laboratory (DUSEL) Raby et al. (2008) experiments Li et al. (2010b, 2011c). The lightest CP-even Higgs boson mass can be increased Huo et al. (2011), hybrid inflation can be naturally realized, and the correct cosmic primordial density fluctuations can be generated Kyae and Shafi (2006).
Vi No-Scale Foundations of -
In the traditional framework, supersymmetry is broken in the hidden sector, and then its breaking effects are mediated to the observable sector via gravity or gauge interactions. In GUTs with gravity mediated supersymmetry breaking, also known as the minimal Supergravity (mSUGRA) model, the supersymmetry breaking soft terms can be parameterized by four universal parameters: the gaugino mass , scalar mass , trilinear soft term , and the ratio of Higgs VEVs at low energy, plus the sign of the Higgs bilinear mass term . The term and its bilinear soft term are determined by the -boson mass and after the electroweak (EW) symmetry breaking.
To solve the cosmological constant problem, No-Scale Supergravity was proposed Cremmer et al. (1983); Ellis et al. (1984a, b, c); Lahanas and Nanopoulos (1987). No-scale Supergravity is defined as the subset of Supergravity models which satisfy the following three constraints Cremmer et al. (1983); Ellis et al. (1984a, b, c); Lahanas and Nanopoulos (1987): (i) the vacuum energy vanishes automatically due to the suitable Kähler potential; (ii) at the minimum of the scalar potential, there are flat directions which leave the gravitino mass undetermined; (iii) the super-trace quantity is zero at the minimum. Without this, the large one-loop corrections would force to be either zero or of Planck scale. A simple Kähler potential which satisfies the first two conditions is
where is a modulus field and are matter fields. The third condition is model dependent and can always be satisfied in principle Ferrara et al. (1994).
The scalar fields of Eq. (3) parameterize the coset space , where is the number of matter fields. Analogous structures appear in the extended Supergravity theories Cremmer and Julia (1979), for example, for , which can be realized in the compactifications of string theory Witten (1985); Li et al. (1997). The non-compact structure of the symmetry implies that the potential is not only constant but actually identical to zero. For the simple example Kähler potential given above, one can readily check that the scalar potential is automatically positive semi-definite, and has a flat direction along the field. Likewise, it may be verified that the simplest No-Scale boundary conditions emerge dynamically, while may be non-zero at the unification scale, allowing for low energy SUSY breaking.
The specific Kähler potential of Eq. (3) has been independently derived in both weakly coupled heterotic string theory Witten (1985) and the leading order compactification of M-theory on Li et al. (1997). Note that in both cases, the Yang-Mills fields span a ten dimensional space-time. It is not obtained directly out of F-theory, as represented for example by the strong coupling lift from Type IIB intersecting D-brane model building with D7- and D3-branes Beasley et al. (2009a, b); Donagi and Wijnholt (2008a, b), where the Yang-Mills fields on the D7-branes occupy an eight dimensional space-time. Nevertheless, it is certainly possible in principle to calculate a gauge kinetic function, Kahler potential and superpotential in the context of Type IIB interecting D-brane model building, and the F-theory could thus admit a more general definition of No-Scale Supergravity, as realized by a Kähler potential like
where only three of the moduli fields and may yield non-zero F-terms.
In Ref. Giddings et al. (2002), No-Scale Supergravity was obtained in the Type IIB and F-theory compactifications at the leading order. Likewise, the subsequently introduced KKLT Kachru et al. (2003) constructions also manifest a No-Scale SUGRA structure at the classical level. Indeed, the No-Scale features are generically obtained at the tree-level in string theory compactifications due to the presence of three complex extra dimensions. However, this classical level result is rather precariously balanced, and may be spoiled by quantum corrections to the superpotential including flux contributions, instanton effects, gaugino condensation, and the next order corrections. In this sense, we consider the KKLT type SUGRA models as a generalization or extension of the elemental No-Scale form.
The No-Scale - model under discussion has been constructed locally in F-theory Jiang et al. (2009, 2010), although the mass of the additional vector-like multiplets, and even the fact of their existence, is not mandated by the F-theory, wherein it is also possible to realize models with only the traditional Flipped (or Standard) field content. We claim only an inherent consistency of their conceptual origin out of the F-theoretic construction, and take the manifest phenomenological benefits which accompany the natural elevation of the secondary GUT unification phase to GeV as justification for the greater esteem which we hold for this particular model above other alternatives. There are, though, also delicate questions of compatibility between the local F-theoretic model building origins and the purely field-theoretic RGE running which we employ up to the presumed high scale. As one approaches the Planck mass , consideration must be given to the role which will be played by Kaluza Klein (KK) and string mode excitations, and also to corrections of order from stabilization of the global volume of the six-dimensional internal space in association with the establishment of the string scale .
The most important question is whether our model can in fact be embedded into a globally consistent framework. Without such, we do not know the concrete Kähler potential of the SM fermions and Higgs fields, and cannot by this means explicitly calculate the supersymmetry breaking scalar masses and trilinear soft terms. This construction remains elusive though, and is beyond the reach of the current work. Regardless, one may anticipate that in such a globally consistent model, a string scale of order GeV would indeed be realized, as in the weakly coupled heterotic string theory, tying in nicely with our naïvely projected value for . It seems additionally that a field-theoretic application of the No-Scale boundary conditions might prove to be validated in this case. Moreover, we would not necessarily require the presence of instanton effects or gaugino condensation for stabilization of the modulus as in the KKLT mechanism. This is crucial, because such effects can have the negative side effect of destroying the leading No-Scale structure. In fact, we could have no gaugino condensation at all, or the superpotential from gaugino condensation might only depend on , as again exemplified in the Type IIB intersecting D-brane models Blumenhagen et al. (2005).
Such considerations, coupled with the demonstrated testability and phenomenological success of the first order analysis in the simplest No-Scale SUGRA framework, argue for a continuing study of the generalized No-Scale SUGRA picture. It is important to note that there exist several such generalizations, including the previously mentioned Type II intersecting D-brane models Cvetic et al. (2004); Chen et al. (2008a, b), mirage mediation of flux compactifications Choi et al. (2004, 2005), and the extraction of SUSY breaking soft terms from the leading order compactification of M-theory on Nilles et al. (1997, 1998); Lukas et al. (1998, 1999); Li (1999); in the latter case we have previously obtained (in a different model context) a generalization employing modulus dominated SUSY breaking Li (1999).
In this paper, however, we maintain a “first steps first” perspective, concentrating on the simplest No-Scale Supergravity and reserving any such extensions for the future. The potential for stringy modifications duly noted, we then essentially aim to study an F-theory inspired variety of low energy SUSY phenomenology, remaining agnostic as to the details of the Kähler structure. Nevertheless, by studying the simplest No-Scale Supergravity, we may still expect to encapsulate the correct leading order behavior. We likewise maintain the simplicity of a leading order approximation by neglecting consideration of any stringy threshold corrections, the substantive onset of which is anyway expected to be deferred to , the true GUT scale of this model. It should be added that since the running of the gauge couplings is logarithmically dependent upon the mass scale, the contributions to the RGEs from the string and KK mode excitations are quite small.
Vii The Super No-Scale Mechanism
The single relevant modulus field in the simplest stringy No-Scale Supergravity is the Kähler modulus , a characteristic of the Calabi-Yau manifold, the dilaton coupling being irrelevant. We consider the gaugino mass as a useful modulus related to the F-term of , stipulating, in other words, that the gauge kinetic function must depend on . This is realized, for example, in the Type IIB intersecting D-brane models Blumenhagen et al. (2005) where gauge kinetic functions explicitly depend on both and , as in Eq. (4). Again, since the F-theory may be considered as a strongly coupled formulation of the Type IIB string theory, it is natural to believe that the gauge kinetic function under this lift depends on as well. While the limit is quite suggestive, lacking still a concrete globally consistent embedding, we cannot definitively prove that the superpotential remains unperturbed by .
Proceeding tentatively as such, the F-term of generates the gravitino mass , which is proportionally equivalent to . Exploiting the simplest No-Scale boundary condition at and running from high energy to low energy under the RGEs, there can be a secondary minimization, or minimum minimorum, of the minimum of the Higgs potential for the EWSB vacuum. Since depends on , the gaugino mass is consequently dynamically determined by the equation , aptly referred to as the “Super No-Scale” mechanism Li et al. (2010a).
It could easily have been that in consideration of the above technique, there were: A) too few undetermined parameters, with the condition forming an incompatible over-constraint, and thus demonstrably false, or B) so many undetermined parameters that the dynamic determination possessed many distinct solutions, or was so far separated from experiment that it could not possibly be demonstrated to be true. The actual state of affairs is much more propitious, being specifically as follows. The three parameters are once again identically zero at the boundary because of the defining Kähler potential, and are thus known at all other scales as well by the RGEs. The minimization of the Higgs scalar potential with respect to the neutral elements of both SUSY Higgs doublets gives two conditions, the first of which fixes the magnitude of . The second condition, which would traditionally be used to fix , instead here enforces a consistency relationship on the remaining parameters, being that is already constrained.
In general, the condition gives a hypersurface of solutions cut out from a very large parameter space. If we lock all but one parameter, it will give the final value. If we take a slice of two dimensional space, as has been described, it will give a relation between two parameters for all others fixed. In a three-dimensional view with on the vertical axis, this curve is the “flat direction” line along the bottom of the trench of solutions. In general, we must vary at least two parameters rather than just one in isolation, in order that their mutual compensation may transport the solution along this curve. The most natural first choice is in some sense the pair of prominent unknown inputs and , as was demonstrated in Ref. Li et al. (2010a).
Having come to this point, it is by no means guaranteed that the potential will form a stable minimum. It must be emphasized that the No-Scale boundary condition is the central agent affording this determination, as it is the extraction of the parameterized parabolic curve of solutions in the two compensating variables which allows for a localized, bound nadir point to be isolated by the Super No-Scale condition, dynamically determining both parameters. The background surface of for the full parameter space outside the viable subset is, in contrast, a steadily inclined and uninteresting function. In our prior study, the local minimum minimorum of for the choices GeV and GeV dynamically established , and . Although we have remarked that and have no directly established experimental values, they are severely indirectly constrained by phenomenology in the context of this model Li et al. (2011a, b). It is highly non-trivial that there should be accord between the top-down and bottom-up perspectives, but this is indeed precisely what has been observed Li et al. (2010a).
Viii The GUT Higgs Modulus
An alternate pair of parameters for which one may attempt to isolate a curve, which we consider for the first time in this work, is that of and the GUT scale , at which the and couplings initially meet. Fundamentally, the latter corresponds to the modulus which sets the total magnitude of the GUT Higgs field’s VEVs. could of course in some sense be considered a “known” quantity, taking the low energy couplings as input. Indeed, starting from the measured SM gauge couplings and fermion Yukawa couplings at the standard GeV electroweak scale, we may calculate both and the final unification scale , and subsequently the unified gauge coupling and SM fermion Yukawa couplings at , via running of the RGEs. However, since the VEVs of the GUT Higgs fields and are considered here as free parameters, the GUT scale must not be fixed either. As a consequence, the low energy SM gauge couplings, and in particular the gauge coupling , will also run freely via this feedback from .
We consider this conceptual release of a known quantity, in order to establish the nature of the model’s dependence upon it, to be a valid and valuable technique, and have employed it previously with specific regards to “postdiction” of the top quark mass value Li et al. (2011b). Indeed, forcing the theoretical output of such a parameter is only possible in a model with highly constrained physics, and it may be expected to meet success only by intervention of either grand coincidence or grand conspiracy of Nature. Simultaneous to the recognition of the presence of a second dynamic modulus, we lock down the value of , which by contrast is a simple numerical parameter, and ought then to be treated in a manner consistent with the top quark and vector-like mass parameters. For this study, we choose a vector-like particle mass GeV, and use the experimental top quark mass input GeV. We emphasize that the choice of is not an arbitrary one, since a prior analysis Li et al. (2011b) has shown that a TeV vector-like mass is in compliance with all current experimental data and the No-Scale =0 requirement. The constant parameter is set consistent with its value prior to the variation of the GUT modulus.
In actual practice, the variation of is achieved in the reverse by programmatic variation of the Weinberg angle, holding the strong and electromagnetic couplings at their physically measured values. Figure 1 demonstrates the scaling between , (logarithmic axis), and the -boson mass. The variation of is attributed primarily to the motion of the electroweak couplings, the magnitude of the Higgs VEV being held essentially constant. We ensure also that the unified gauge coupling, SM fermion Yukawa couplings, and specifically also the Higgs bilinear term GeV, are each held stable at the scale to correctly mimic the previously described procedure.
The parameter ranges for the variation depicted in Fig. 1 are , , and GeV, and likewise also the same for Figs. (2-8), which will feature subsequently. The minimum minimorum falls at the boundary of the prior list, dynamically fixing GeV and placing again in the vicinity of GeV. The low energy SM gauge couplings are simultaneously constrained by means of the associated Weinberg angle, with , in excellent agreement with experiment. The corresponding range of predicted proton lifetimes in the leading modes is years Li et al. (2011c). If the GUT scale becomes excessively light, below about GeV, then proton decay would be more rapid than allowed by the recently updated lower bound of years from Super-Kamiokande Nishino et al. (2009).
We are cautious against making a claim in precisely the same vein for the dynamic determination of GeV, since again the crucial electroweak Higgs VEV is not a substantial element of the variation. However, in conjunction with the radiative electroweak symmetry breaking Ellis et al. (1983b, c) numerically implemented within the SuSpect 2.34 code base Djouadi et al. (2007), the fixing of the Higgs VEV and the determination of the electroweak scale may also plausibly be considered legitimate dynamic output, if one posits the scale input to be available a priori.
By extracting a constant slice of the hyper-surface, the secondary minimization condition on is effectively rotated, albeit quite moderately, relative to the procedure of Ref. (Li et al. (2010a)). The present minimization, referencing , and , is again dependent upon and , while the previously described Li et al. (2010a) determination of was, by contrast, and invariant. Recognizing that a minimization with all three parameters simultaneously active is required to declare all three parameters to have been simultaneously dynamically determined, we emphasize the mutual consistency of the results. We again stress that the new minimum minimorum is also consistent with the previously advertised golden strip, satisfying all presently known experimental constraints to our available resolution. It moreover also addresses the problems of the SUSY breaking scale and gauge hierarchy Li et al. (2010a), insomuch as is determined dynamically.
Ix The Minimum Minimorum of the Electroweak Higgs Potential
In supersymmetric SMs, there is a pair of Higgs doublets and which give mass to the up-type quarks and down-type quarks/charged leptons, respectively. The one-loop effective Higgs potential in the ’t Hooft-Landau gauge and in the scheme is given by
where and are the supersymmetry breaking soft masses, and are respectively the gauge couplings of and , and are respectively the degree of freedom and mass for , and is the renormalization scale. In our numerical results in the figures, we shall designate differences in the fourth-root of the effective Higgs potential as , measured in units of GeV relative to an arbitrary overall zero-offset.
We have revised the SuSpect 2.34 code base Djouadi et al. (2007) to incorporate our specialized No-Scale - with vector-like mass algorithm, and accordingly employ two-loop RGE running for the SM gauge couplings, and one-loop RGE running for the SM fermion Yukawa couplings, term, and SUSY breaking soft terms. For our choice of GeV, GeV, and GeV, we present the one-loop effective Higgs potential in terms of and in Fig. 2, in terms of and in Fig. 3, in terms of and tan in Fig. 4, and in terms of and in Fig. 5, where , , and . These figures clearly demonstrate the localization of the minimum minimorum of the Higgs potential, corroborating the dynamical determination of and GeV in Li et al. (2010a).
Additionally, we exhibit the space in Fig. 6, the space in Fig. 7, and the space in Fig. 8, where . Fig. 6 demonstrates that GeV at the minimum minimorum, which correlates to GeV, or more directly, . Together, the alternate perspectives of Figs. 6, 7, and 8 complete the view given in Figs. 2, 3, 4, and 5 to visually tell the story of the dynamic interrelation between the , , and scales, as well as the electroweak gauge couplings, and the Higgs VEVs. The curves in each of these figures represent only those points that satisfy the = 0 requirement, as dictated by No-Scale Supergravity, serving as a crucial constraint on the dynamically determined parameter space. Ultimately, it is the significance of the requirement that separates the No-Scale - with vector-like particles from the entire compilation of prospective string theory derived models. By means of the = 0 vehicle, No-Scale - has surmounted the paramount challenge of phenomenology, that of dynamically determining the electroweak scale, the scale of fundamental prominence in particle physics.
We wish to note that recent progress has been made in incorporating more precise numerical calculations into our baseline algorithm for No-Scale - with vector-like particles. Initially, when we commenced the task of fully developing the phenomenology of this model, the extreme complexity of properly numerically implementing No-Scale - with vector-like particles compelled a gradual strategy for construction and persistent enhancement of the algorithm. Preliminary findings of a precision improved algorithm indicate that compliance with the 7-year WMAP relic density constraints requires a slight upward shift to from the value computed in Ref. Li et al. (2011a), suggesting a potential convergence to even finer resolution of the dynamical determination of given by the Super No-Scale mechanism, and the value demanded by the experimental relic density measurements. We shall furnish a comprehensive analysis of the precision improved algorithm at a later date.
X Probing The Blueprints of the No-Scale Multiverse at the Colliders
We offer in closing a brief summary of direct collider, detector, and telescope level tests which may probe the blueprints of the No-Scale Multiverse which we have laid out. As to the deep question of whether the ensemble be literal in manifestation, or merely the conceptual superset of unrealized possibilities of a single island Universe, we pretend no definitive answer. However, we have argued that the emergence ex nihilo of seedling universes which fuel an eternal chaotic inflation scenario is particularly plausible, and even natural, within No-Scale Supergravity, and our goal of probing the specific features of our own Universe which might implicate its origins in this construction are immediately realizable and practicable.
The unified gaugino at the unification scale can be reconstructed from impending LHC events by determining the gauginos , , and at the electroweak scale, which will in turn require knowledge of the masses for the neutralinos, charginos, and the gluino. Likewise, can be ascertained in principle from a distinctive experimental observable, as was accomplished for mSUGRA in Arnowitt et al. (2008). We will not undertake a comprehensive analysis here of the reconstruction of and , but will offer for now a cursory examination of typical events expected at the LHC. We leave the detailed compilation of the experimental observables necessary for validation of the No-Scale - at the LHC for the future, and we especially encourage those specializing in such research to investigate the No-Scale -.
For the benchmark SUSY spectrum presented in Table 1, we have adopted the specific values , and . We expect that higher order corrections will shift the precise location of the minimum minimorum a little bit, for example, within the encircled gold-tipped regions of the diagrams in the prior section. We have selected a ratio for at the lower end of this range for consistency with our previous study Li et al. (2010a), and to avoid stau dark matter.
At the benchmark point, we calculate for the cold dark matter relic density. The phenomenology is moreover consistent with the LEP limit on the lightest CP-even Higgs boson mass, GeV Barate et al. (2003, 2003), the CDMSII Ahmed et al. (2009) and Xenon100 Aprile et al. (2010) upper limits on the spin-independent cross section , and the Fermi-LAT space telescope constraints Abdo et al. (2010) on the photon-photon annihilation cross section . The differential cross-sections and branching ratios have been calculated with PGS4 Conway et al. (2009) executing a call to PYTHIA 6.411 Sjostrand et al. (2006), using our specialized No-Scale algorithm integrated into the SuSpect 2.34 code for initial computation of the sparticle masses.
The benchmark point resides in the region of the experimentally allowed parameter space that generates the relic density through stau-neutralino coannihilation. Hence, the five lightest sparticles for this benchmark point are . Here, the gluino is lighter than all the squarks with the exception of the lightest stop, so all squarks will predominantly decay to a gluino and hadronic jet, with a small percentage of squarks producing a jet and either a or . The gluinos will decay via virtual (off-shell) squarks to neutralinos or charginos plus quarks, which will further cascade in their decay. The result is a low-energy tau through the processes and .
The LHC final states of low-energy tau in the - stau-neutralino coannihilation region are similar to those same low-energy LHC final states in mSUGRA, however, in the stau-neutralino coannihilation region of mSUGRA, the gluino is typically heavier than the squarks. The strong coupling effects from the additional vector-like particles on the gaugino mass RGE running reduce the physical gluino mass below the squark masses in -. As a consequence, the LHC final low-energy tau states in the stau-neutralino coannihilation regions of - and mSUGRA will differ in that in -, the low-energy tau states will result largely from neutralinos and charginos produced by gluinos, as opposed to the low-energy tau states in mSUGRA resulting primarily from neutralinos and charginos produced from squarks.
Also notably, the TeV-scale vector-like multiplets are well targeted for observation by the LHC. We have argued Li et al. (2011b) that the eminently feasible near-term detectability of these hypothetical fields in collider experiments, coupled with the distinctive flipped charge assignments within the multiplet structure, represents a smoking gun signature for Flipped , and have thus coined the term flippons to collectively describe them. Immediately, our curiosity is piqued by the recent announcement Abazov et al. (2011) of the DØ collaboration that vector-like quarks have been excluded up to a bound of 693 GeV, corresponding to the immediate lower edge of our anticipated range for their discovery Li et al. (2011b).
The advancement of human scientific knowledge and technology is replete with instances of science fiction transitioning to scientific theory and eventually scientific fact. The conceptual notion of a “Multiverse” has long fascinated the human imagination, though this speculation has been largely devoid of a substantive underpinning in physical theory. The modern perspective presented here offers a tangible foundation upon which legitimate discussion and theoretical advancement of the Multiverse may commence, including the prescription of specific experimental tests which could either falsify or enhance the viability of our proposal. Our perspective diverges from the common appeals to statistics and the anthropic principle, suggesting instead that we may seek to establish the character of the master theory, of which our Universe is an isolated vacuum condensation, based on specific observed properties of our own physics which might be reasonably inferred to represent invariant common characteristics of all possible universes. We have focused on the discovery of a model universe consonant with our observable phenomenology, presenting it as confirmation of a non-zero probability of our own Universe transpiring within the larger String Landscape.
The archetype model universe which we advance in this work implicates No-Scale Supergravity as the ubiquitous supporting structure which pervades the vacua of the Multiverse, being the crucial ingredient in the emanation of a cosmologically flat universe from the quantum “nothingness”. In particular, the model dubbed No-Scale - has demonstrated remarkable consistency between parameters determined dynamically (the top-down approach) and parameters determined through the application of current experimental constraints (the bottom-up approach). This enticing convergence of theory with experiment elevates No-Scale -, in our estimation, to a position as the current leading GUT candidate. The longer term viability of this suggestion is likely to be greatly clarified in the next few years, based upon the wealth of forthcoming experimental data.
Building on the results presented in prior works Li et al. (2011a, b, 2010a), we have presented a dynamic determination of the penultimate Flipped unification scale , or more fundamentally, the GUT Higgs VEV moduli. We have demonstrated that the = 0 No-Scale boundary condition is again vital in dynamically determining the model parameters. Procedurally, we have fixed the unified gauge coupling, SM fermion Yukawa couplings, and Higgs bilinear term at the final unification scale , while concurrently allowing the VEVs of the GUT Higgs fields and to float freely, as driven by and the low energy SM gauge couplings, via variation of the Weinberg angle. Employing the “Super No-Scale” condition to secondarily minimize the effective Higgs potential, we have obtained GeV, , and .
The blueprints which we have outlined here, integrating precision phenomenology with prevailing experimental data and a fresh interpretation of the Multiverse and the Landscape of String vacua, offer a logically connected point of view from which additional investigation may be mounted. As we anticipate the impending stream of new experimental data which is likely to be revealed in ensuing years, we look forward to serious discussion and investigation of the perspective presented in this work. Though the mind boggles to contemplate the implications of this speculation, so it must also reel at even the undisputed realities of the Universe, these acknowledged facts alone being manifestly sufficient to humble our provincial notions of longevity, extent, and largess.
This research was supported in part by the DOE grant DE-FG03-95-Er-40917 (TL and DVN), by the Natural Science Foundation of China under grant No. 10821504 (TL), and by the Mitchell-Heep Chair in High Energy Physics (TL).
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