GUT Precursors and Entwined SUSY:The Phenomenology of Stable Non-Supersymmetric Strings

GUT Precursors and Entwined SUSY: The Phenomenology of Stable Non-Supersymmetric Strings

Abstract

Recent work has established a method of constructing non-supersymmetric string models that are stable, with near-vanishing one-loop dilaton tadpoles and cosmological constants. This opens up the tantalizing possibility of realizing stable string models whose low-energy limits directly resemble the Standard Model rather than one of its supersymmetric extensions. In this paper we consider the general structure of such strings and find that they share two important phenomenological properties. The first is a so-called “GUT-precursor” structure in which new GUT-like states appear with masses that can be many orders of magnitude lighter than the scale of gauge coupling unification. These states allow a parametrically large compactification volume, even in weakly coupled heterotic strings, and in certain regions of parameter space can give rise to dramatic collider signatures which serve as “smoking guns” for this overall string framework. The second is a residual “entwined-SUSY” (or e-SUSY) structure for the matter multiplets in which different multiplet components carry different horizontal charges. As a concrete example and existence proof of these features, we present a heterotic string model that contains the fundamental building blocks of the Standard Model such as the Standard-Model gauge group, complete chiral generations, and Higgs fields — all without supersymmetry. Even though massless gravitinos and gauginos are absent from the spectrum, we confirm that this model has an exponentially suppressed one-loop dilaton tadpole and displays both the GUT-precursor and e-SUSY structures. We also discuss some general phenomenological properties of e-SUSY, such as cancellations in radiative corrections to scalar masses, the possible existence of a corresponding approximate moduli space, and the prevention of rapid proton decay.

I Introduction

Most approaches to string phenomenology have historically proceeded under the assumption that the Standard Model (SM) ultimately becomes supersymmetric at a higher energy scale parametrically near the electroweak symmetry-breaking scale. One then attempts to realize the resulting supersymmetric theory as the low-energy limit of a supersymmetric string. This approach was motivated by many factors. While bottom-up factors included a strong belief in the existence of weak-scale supersymmetry, a critical top-down factor was the fact that non-supersymmetric strings are generally unstable, with large one-loop dilaton tadpoles. The existence of such tadpoles destabilizes these strings, and thus renders them inconsistent in a way that does not arise for supersymmetric strings.

In recent work Abel:2015oxa (), we advocated a new approach to this problem. Specifically, even though non-supersymmetric strings are generally unstable, they may nevertheless be metastable — i.e., endowed with lifetimes that are large compared with the age of the universe. Indeed this metastability can be arranged not through the existence of a potential barrier through which an eventual non-perturbative tunneling might occur, but simply by having one-loop dilaton tadpoles whose values — although non-zero — are exponentially suppressed. Thus, while such strings do not necessarily sit at true minima of the dilaton potential, the potential slopes that they experience are exponentially suppressed. Such strings therefore remain effectively stable at their original locations for all relevant cosmological timescales.

In Ref. Abel:2015oxa (), we demonstrated how such metastable strings may be constructed within the perturbative heterotic framework. Moreover, as we demonstrated, the low-energy limits of these strings may even resemble the Standard Model or one of its grand-unified extensions Abel:2015oxa (). This then opens up the possibility of developing a fully non-supersymmetric string phenomenology — one in which the Standard Model itself is realized directly as the low-energy limit of a non-supersymmetric string. Indeed, such models take the general form of a low-energy theory in which supersymmetry (SUSY) is broken at arbitrarily high scales, yet with a one-loop cosmological constant and dilaton tadpole that are exponentially suppressed — all capped off with a self-consistent ultraviolet (UV) completion which is entirely non-supersymmetric. Indeed, as discussed in Ref. Abel:2016hgy (), although these theories admit low-energy descriptions in terms of four-dimensional effective field theories with broken supersymmetry, they are never even approximately supersymmetric in four dimensions.

In this paper, we take the next steps in exploring the phenomenological implications of this approach. In particular, because our construction necessarily involves large-volume compactifications, one pressing issue concerns the behavior of the gauge couplings — especially if we require perturbativity both at the electroweak scale as well as in the UV limit. As we shall discuss, this requires that our strings exhibit a variant of the so-called “GUT precursor” structure originally proposed in Refs. Dienes:2002bg (); Dienes:2004rt ().  Tightly coupled with this, we shall also argue that the (chiral) matter fields of such strings exhibit a so-called “entwined SUSY” or e-SUSY in which these states and their would-be superpartners have different charges under a horizontal symmetry. This horizontal symmetry is thus non-trivially “entwined” with the same physics that renders the theory non-supersymmetric and also breaks the GUT symmetry. In this connection, we note that entwined SUSY is reminiscent of the so-called “folded SUSY” framework Burdman:2006tz (); Cohen:2015gaa () in which would-be superpartners have different charges. Indeed, it might even be possible to incorporate folded SUSY or its variants into our construction. However, as we shall see, it is actually entwined SUSY which unavoidably emerges from our overall stable-string construction and which even serves as one its predictions.

The construction of non-supersymmetric strings has been explored by a number of authors in recent years (see, for example, Refs. Angelantonj:2014dia (); Hamada:2015ria (); Nibbelink:2015ena (); Florakis:2015txa (); Ashfaque:2015vta (); Blaszczyk:2015zta (); Nibbelink:2015vha (); Angelantonj:2015nfa (); Satoh:2015nlc (); Athanasopoulos:2016aws (); Sugawara:2016lpa (); Kounnas:2016gmz (); Florakis:2016ani (); Satoh:2016izo (); Kounnas:2017mad (); Florakis:2017ecd (); Faraggi:2017cnh (); Coudarchet:2017pie (); Mourad:2017rrl ()). This growing literature indicates an increasing interest in this subject, presumably motivated not only by the apparent experimental absence of supersymmetry at the Large Hadron Collider but also by the intrinsically different theoretical behavior of strings within this hitherto largely unexplored region of the string landscape. However, within this literature, what distinguishes our work is its focus on the fundamental stability properties of such strings, at least as far as their dilaton tadpoles are concerned. Indeed, the presence of a non-zero dilaton tadpole indicates that the fundamental string vacuum is unstable. It is thus only by concentrating on string models with vanishing or near-vanishing dilaton tadpoles that one can be assured of working in string vacua whose stability properties resemble those of their supersymmetric cousins. Of course, just as for supersymmetric strings, there will always remain further moduli which also require stabilization through either string-theoretic or field-theoretic means. However, we view the dilaton tadpole as uniquely problematic in the construction of non-supersymmetric strings, as the existence of such a tadpole is the direct hallmark of the breaking of supersymmetry. This problem must therefore be tackled at the outset. Indeed, it is only after the effective cancellation of this tadpole that we can proceed to consider the development of a non-supersymmetric string phenomenology on a par with that of strings with spacetime supersymmetry.

This paper is organized as follows. First, in Sect. II, we review our general framework Abel:2015oxa () for the construction of non-supersymmetric heterotic strings with exponentially suppressed one-loop dilaton tadpoles. Then, in Sect. III, we discuss how and why these strings inevitably give rise to not only GUT precursors but also an entwined SUSY — observations that form the central core of this paper. In Sect. IV we then proceed to construct a self-consistent non-supersymmetric heterotic string model which exhibits all of these properties. Our aim is to present not only a concrete example and existence proof of these features within the context of a fully self-consistent string model, but also to demonstrate that these features can coexist with other fundamental phenomenological building blocks of realistic string models such as the Standard-Model gauge group, complete chiral generations, and Higgs fields — all in a stable, non-supersymmetric setting. In Sect. V, we then briefly discuss several other phenomenological aspects of metastable string models that result from their GUT-precursor and e-SUSY structures. These include cancellations in radiative corrections to scalar masses, the possible existence of a corresponding approximate moduli space, and the prevention of rapid proton decay. Finally, in Sect. VI, we discuss a variety of open topics and future directions related to our work. Details pertaining to a calculation in Sect. IV are collected in an Appendix.

Ii Stable non-supersymmetric strings:  Basic framework

We begin by briefly summarizing the framework described in Ref. Abel:2015oxa () for constructing closed, non-supersymmetric string theories with exponentially suppressed one-loop dilaton tadpoles. All of the strings we consider in this paper will be members of this class.

There are two critical features which define this class of models. First, these models are all what may be called “interpolating” models. Specifically, each is a compactification of a higher-dimensional string model , and as such is endowed with an adjustable compactification volume . As , we reproduce the original uncompactified string model . However, as , we are assured by T-duality that we produce a string model which may be considered to be the T-dual of another higher-dimensional model . If the compactification is untwisted, then will be nothing other than . However, if the compactification is twisted, then will generally differ from . In such cases, we can view our compactified model as smoothly “interpolating” between the uncompactified models (as ) and (as ). Note that the requirement that both and be bona-fide self-consistent string models provides a set of tight constraints on the twists which may be applied when compactifying  BlumDienes (); DienesLennekSharma (); Abel:2015oxa (); Aaronson ().

The second feature that defines this class of models has to do with the choices of and . Certain requirements for these choices are relatively straightforward: for example, we will require and to be supersymmetric and non-supersymmetric, respectively. This guarantees that the endpoint of the interpolation has a vanishing one-loop tadpole but that our interpolating model is otherwise non-supersymmetric for all finite . This also provides us with an “order parameter” for dialing the degree of supersymmetry breaking. However, other requirements for our choices of and are less straightforward. In particular, for any given choice of , only certain choices for (or equivalently only certain choices of the SUSY-breaking twist that will be introduced into the compactification) are suitable for generating the desired exponentially suppressed dilaton tadpole for large , even if is only moderately large. Specifically, we must choose so that this twist leaves an equal number of massless bosonic and fermionic degrees of freedom in the spectrum of the resulting interpolating string model. In other words, even though this twist breaks spacetime supersymmetry (so that the resulting string spectrum contains no massless gravitinos, for example), it must be carefully chosen so that the resulting spectrum nevertheless exhibits an equal number of massless bosonic and fermionic degrees of freedom. Note, in particular, that there need be no other relation between the bosonic and fermionic degrees of freedom. For example, these degrees of freedom can carry entirely different gauge charges, with a gluon degree of freedom balanced against a neutrino degree of freedom. Likewise, some of these degrees of freedom can reside in a visible sector while others reside in a hidden sector. Thus we need not even have equal numbers of massless bosonic and fermionic degrees of freedom in each sector separately. All that matters are the total numbers of massless degrees of freedom, summed over all sectors of the theory.

For any given string model , it is not guaranteed that there exists a suitable model that will produce an interpolating model within this class. In other words, for any given model , there may not necessarily exist a suitable twist that can be introduced upon compactification which yields a non-supersymmetric interpolating model with boson/fermion degeneracy at the massless level. For this reason, the art of choosing suitable models and can be quite intricate, and methods for this purpose are described in Ref. Abel:2015oxa ().  But what is remarkable is that these are the only requirements for building metastable string models. Once and are chosen satisfying these properties, a unique interpolating model is determined which will be a member of the desired class.

Figure 1: The spectrum of a generic metastable interpolating model for . States with masses below (i.e., below ) consist of massless observable states, massless hidden-sector states, their would-be superpartners, and their lightest KK excitations. For these lightest states, the net (bosonic minus fermionic) numbers of degrees of freedom from the hidden sector are exactly equal and opposite, level by level, to those from the observable sector. This is true for all large compactification radii. Note that this cancellation of net physical-state degeneracies between the observable and hidden sectors bears no connection with any supersymmetry, either exact or approximate, in the string spectrum. For the heavier states, by contrast, the observable and hidden sectors need no longer supply equal and opposite numbers of degrees of freedom. The properties of these sectors are nevertheless governed by misaligned-supersymmetry constraints, as a result of which the entire string spectrum continues to satisfy the supertrace relations in Eq. (1). These relations maintain the finiteness of the overall string theory, even without spacetime supersymmetry. Figure taken from Ref. Abel:2015oxa ().

Because the breaking of supersymmetry in this framework is tied to the compactification, what results is an interpolating model whose spectrum has certain characteristic features for large compactification volume (i.e., for , where denotes the dimensionality of and where the symbol ’’ denotes a factor of or more). The generic spectrum of such string models is sketched in Fig. 1. Situated at the massless level are states that together have equal numbers of bosonic and fermionic degrees of freedom. However the would-be superpartners of these states are no longer massless, but instead have masses . This reflects the breaking of spacetime supersymmetry, leading us to a rough identification of as the scale of supersymmetry-breaking. However, in this context it is important to stress that the massless states by themselves must have equal numbers of bosonic and fermionic degrees of freedom; note in particular that this is not a residual supersymmetric pairing of massless states with their would-be superpartners. However, because the states with masses are the would-be superpartners of the massless states, they too will exhibit equal numbers of bosonic and fermionic degrees of freedom amongst themselves. Note, also, that the states at each mass level may be arbitrarily split between observable and hidden sectors, as mentioned above and indicated in Fig. 1.  Consequently the equalities between the numbers of bosonic and fermionic degrees of freedom amongst the light states in these string models need not be observable in any way.

Proceeding further upwards in mass then leads to a whole spectrum of repeating Kaluza-Klein (KK) excitations which echo this basic structure, so that the first KK excitations of the massless states have masses while the first KK excitations of the would-be superpartners have masses . This structure is then replicated at regular mass intervals . Each of these levels therefore continues to exhibit equal numbers of bosonic and fermionic degrees of freedom, even though no supersymmetry is present. Ultimately, however, we reach the mass scale at which the first string excitations appear. In general, states with non-zero string excitation numbers have masses . Unlike the massless states, however, the states with need no longer come with equal numbers of bosonic and fermionic degrees of freedom at each mass level. Thus, for masses , the equality between bosonic and fermionic degrees of freedom is lost. These states nevertheless exhibit a residual property called “misaligned supersymmetry” missusy (); supertraces () which tightly controls the balancing between bosonic and fermionic degrees of freedom at all mass levels throughout the infinite towers of massless and massive states, and which ensures the ultimate finiteness for which string theory is famous — even without supersymmetry. Indeed, misaligned SUSY is a general property of the spectra of all closed, tachyon-free, non-supersymmetric string models, and for strings within our class guarantees that the bosonic and fermionic states are arranged in such a way that the ordinary supertrace relations

(1)

nevertheless continue to hold at tree level when the summation is over all of the physical (i.e., level-matched) states in the spectrum supertraces (). We shall discuss the precise value of for these strings below.

We are interested in this class of models because of their remarkable stability properties. In general, for a given string model in uncompactified dimensions, the dilaton tadpole is proportional to the one-loop vacuum amplitude (or energy density)

(2)

where is the string partition function and where is the fundamental domain of the modular group. Note that we have expressed in units of , where is the reduced string scale; thus as defined is a dimensionless quantity, while the full energy density (cosmological constant) for the -dimensional theory is , corresponding to a mass scale . However, for interpolating models within the class described above, we find that is severely suppressed as and indeed even for only moderately large compactification volumes. For example, if we are dealing with a one-dimensional compactification (i.e., a compactification on a twisted circle) of radius , as we find Abel:2015oxa ()

(3)

In this expression, is the spacetime dimension of our interpolating model (so that is the spacetime dimension prior to compactification). Likewise, are the numbers of bosonic and fermionic degrees of freedom in the interpolating model at the string level. The first term in Eq. (3) is the leading contribution from the KK excitations of the massless states, while the remaining terms are the leading contributions from states with non-zero string excitations. If , the first term gives the leading contribution , as expected. However, as long as , the first term vanishes and the resulting dilaton tadpole is exponentially suppressed, with a severe suppression factor of the form . In fact, the true suppression for is even stronger than we have indicated here since the difference tends to oscillate in sign as a function of . This is ultimately a result of the misaligned supersymmetry mentioned above. Thus the exponentially suppressed contributions from the terms with tend to interfere against each other, rendering the sum even more suppressed than any single term.

The result in Eq. (3) is remarkable on a number of levels. In particular, there are two aspects which are particularly surprising. The first is the nature of the terms which can be called “field-theoretic”. To understand this issue, we note that a general string theory with a compactification scale can be described through a sequence of different effective theories at different energies. For energies , the effective theory is a four-dimensional quantum field theory (QFT).  Likewise, for , the effective theory is that of a higher-dimensional QFT.  Indeed, it is only for that our theory becomes truly stringy. The same properties would likewise normally be reflected in the amplitudes of such a theory. However, the construction we have described here has the remarkable property that even though is considerably below , the single condition suffices to eliminate all field-theoretic contributions to so that depends on quantities such as in a completely string-theoretic (rather than field-theoretic) manner. This includes not only the four-dimensional QFT-like contributions to , but even the higher-dimensional QFT-like contributions. Ultimately, this situation arises because our framework has the property that the single condition actually ensures the cancellation of the net (boson minus fermion) degeneracies at each KK level all the way up to the first non-zero string excitation. Thus, within our framework, we see that the appropriately normalized -dimensional energy density receives only two groups of leading contributions in Eq. (3): those which scale directly as and which are therefore essentially those of a (compactified) -dimensional QFT, depending only on , and those which scale exponentially with and which are therefore intrinsically stringy, depending on the excited string-oscillator occupation numbers with in Eq. (3).  Indeed, the absence of other contributions which might have scaled as a higher power of and which would have depended on the configuration of non-zero KK excitations below is the hallmark of this framework. Thus, enforcing the single condition leaves only the terms with string-theoretic suppressions and eliminates the leading field-theoretic contributions entirely.

The second remarkable aspect of the result in Eq. (3) concerns the severity of the exponential suppression that arises after the condition is imposed. Clearly, if the supersymmetry had not been broken (i.e., if we had taken ), we would have found . Thus the non-zero value of indicated in Eq. (3) is ultimately the result of taking small but non-zero, so that the masses of the superpartner states are shifted by an amount . Since the contribution to the string partition function from a given state with mass generally scales as , the total contribution to from any state of mass and its would-be superpartner of mass can be viewed as a summation over pairwise combined contributions of the form for various positive values of . For , each such difference is approximately . Thus, one might expect that the mass shifts between the states in our theory and their would-be superpartners would generate a total contribution to which is suppressed as a power of . However, the set of masses over which such a summation is performed is itself dependent on and becomes dense as . Thus, as , the cancellation between states and their would-be superpartners becomes more complete while the density of such states increases. It is ultimately the interplay between these two effects — along with our condition — which produces the severe “inverted” suppression factor quoted in Eq. (3).

What we obtain, then, is a four-dimensional non-supersymmetric string theory governed by three fundamentally different mass scales. The first is , which governs the splitting between states and their would-be superpartners and which may thus be viewed as the scale of supersymmetry breaking. The second is , which governs the energies associated with the string oscillator excitations and which therefore serves as the scale of the UV completion of the theory. Remarkably, however, when , these two scales conspire to produce a new scale

(4)

which is significantly smaller than either of the two previous scales and which sets the magnitude of the corresponding one-loop cosmological constant (vacuum energy). It is this scale which governs the ultimate tree-level dilaton stability of the theory.

The suppression of this last scale can also be understood geometrically. Because supersymmetry is broken through compactification in our construction, massive string modes need to propagate over the full compactification volume, i.e., over a distance , in order to realize a non-zero . This leads to a Yukawa suppression of the form . This geometric understanding ties in with our alternative explanation above since the inverted suppression factor and the “large-volume” Yukawa picture both arise after Poisson resummation.

Viewed from the perspective of string model-building, however, the result in Eq. (3) is extremely beneficial. As we have already noted, the scale of supersymmetry breaking in this construction can roughly be taken to be , or in the one-dimensional case. However, as long as we ensure that , the dilaton-stability of such a string (i.e., the suppression of the corresponding value of ) is not polynomial in but exponential. We can therefore dial (or more generally our compactification volume ) to any value desired — even to the TeV scale — while nevertheless maintaining the required suppression of the dilaton tadpole and assuring the metastability of the non-supersymmetric string. It is for this fundamental reason that our framework leads to a promising starting point for a non-supersymmetric string phenomenology. Furthermore, the value of the supertrace for any tachyon-free closed string theory compactified to four dimensions can be shown supertraces () to scale as , where is defined as in Eq. (2). Thus, the severe suppression of for all string models in this class additionally becomes a suppression for :

(5)

Again, we stress that this occurs even though the scale of SUSY-breaking in this framework is .

When in the following we construct explicit string models, we shall focus on a specific configuration within this general framework which forms a particularly useful testing ground for the more general discussion. As we shall discuss, this configuration is based on perturbative ten-dimensional heterotic strings exhibiting large (GUT-like) gauge symmetries and proceeds through two stages of compactification. The first is a compactification down to dimensions on a manifold or orbifold with volume such that the resulting -dimensional string model is supersymmetric. By contrast, the second stage of compactification from to four dimensions occurs through a -dimensional compactification/interpolation of the type we have been discussing. The space on which this compactification occurs is a -dimensional freely-acting orbifold . We thus have


10D =1 SUSY GUT-like string model (4+)-dimensional =1 SUSY GUT-like string model 4D non-SUSY massless B/F-degeneracy SM-like string model

In Ref. Abel:2015oxa () and in our models to be presented below, we take (so that our intermediate model is six-dimensional) and (with the understanding that the acts on both and ). If this orbifold were untwisted, we would obtain an supersymmetric theory in four dimensions. However, we introduce a Scherk-Schwarz twist which acts not only on the spacetime degrees of freedom but also on the internal gauge degrees of freedom. This coupling between the spacetime twist and the internal gauge twist is ultimately required by modular invariance. We also choose these twists so as to additionally satisfy the conditions laid out above, including the requirement of bose/fermi degeneracy at the massless level. This then produces a non-supersymmetric four-dimensional string model with the desired metastability properties.

In this configuration, both the original GUT-like gauge symmetry and the original spacetime supersymmetry are broken together in the final stage of compactification. It is this feature which ultimately leads to the GUT-precursor and entwined-supersymmetry structures which are the focus of this paper. In fact, we shall even eventually argue in Sect. VI that these structures transcend our particular string construction, and are inevitable within broad classes of non-supersymmetric UV-complete theories. It is therefore to these topics that we now turn.

Iii GUT precursors and entwined SUSY

As indicated above, the starting point of our construction is a higher-dimensional string model exhibiting not only spacetime SUSY but also a GUT gauge symmetry. Both of these symmetries are then broken together upon the final stage of compactification. In principle, the exponential suppression of the dilaton tadpole does not require that we begin with a GUT symmetry prior to compactification. Nor does it require that this symmetry be broken by compactification. Ultimately, these additional features are needed for phenomenological purposes. In this section, we shall begin by explaining why these additional features are needed. We shall then demonstrate that these features inevitably lead to a GUT-precursor structure and the emergence of an entwined supersymmetry. As we shall see, these phenomenological aspects are both quite general and can be understood from a geometric point of view. Indeed, as we shall demonstrate, both entwined SUSY as well as the GUT-precursor structure are rather generic phenomenological properties of a wide class of non-supersymmetric strings — even independently of the need for boson/fermion degeneracy of the massless states. This section thus constitutes the main theoretical portion of this paper, with subsequent sections providing explicit constructions that illustrate these assertions.

iii.1 The problem of gauge couplings in large-volume compactifications

Because our dilaton-stabilization mechanism requires the existence of a large compactification volume, an immediate problem that emerges concerns the values of the gauge and gravitational couplings (sometimes referred to as the “decompactification problem”). It is easy to see how this problem arises. For concreteness, let us consider the case of the heterotic string compactified from ten to four dimensions. In general, the coupling expansion for an -point genus- diagram behaves as

(6)

where is the ten-dimensional string coupling, where is the compactification volume, and where is the (topologically invariant) Euler number of the string worldsheet. Thus at tree level (i.e., for ) the effective four-dimensional Lagrangian for gravitational and gauge interactions scales as and takes the form

(7)

where is the Ricci scalar and is a gauge field strength. From this we can read off the effective four-dimensional tree-level gauge coupling and effective four-dimensional Planck scale :

(8)

where is the compactification volume normalized with respect to the fundamental string scale. From these results it follows that , or equivalently .

The relations in Eq. (8) are completely general. Moreover, for compactification volumes near the string scale [i.e., for ], we find that . The perturbativity condition then requires , and one often chooses in order to make contact with standard logarithmic gauge coupling unification. Indeed, in such a scenario, the measured four-dimensional SM gauge couplings at the weak scale run logarithmically up to the GUT/string scale where they unify into . Note that these gauge couplings run logarithmically over this range precisely because , so that the theory is effectively four-dimensional below . Thus, for , the usual logarithmic gauge coupling unification can be preserved and naturally embedded into string theory Dienespath ().

The situation is very different when the compactification volume is large, as in our configuration. In this case , whereupon the perturbativity condition implies . Such small values for are difficult to reconcile with the measured values of the four-dimensional gauge couplings at the electroweak scale. Of course, the assumption of a large compactification volume implies that , so that our theory is actually higher-dimensional between and . This then opens up an interval over which the running of the gauge couplings above has a power-law (rather than logarithmic) dependence. However, even this observation cannot evade our difficulties. First, with power-law running above , the traditional logarithmic gauge coupling unification is generally lost. Moreover, even though power-law running can still produce a power-law unification of the gauge couplings as one proceeds upwards in energy DDG1 (); DDG2 (); DDG3 (), this unification typically occurs very rapidly after the onset of KK modes, with never very large. Identifying and , we see that this therefore does not leave much room for a large compactification volume . Or, phrased somewhat differently, we might continue to insist that , but this would no longer permit us to parametrically identify with  — a feature that we would generally like to retain.

The question then arises as to how we can reconcile the measured values of the four-dimensional gauge couplings at low energies with an value of the ten-dimensional string coupling  — all in the presence of a large compactification volume , and all while preserving a logarithmic gauge coupling unification at .

iii.2 GUT precursors

It turns out that all of these features are not only reconciled but also realized naturally within the so-called “GUT precursor” scenario originally presented in Refs. Dienes:2002bg (); Dienes:2004rt (). The discussion in Refs. Dienes:2002bg (); Dienes:2004rt () was essentially field-theoretic, but we shall see that this scenario is also a natural prediction of our string framework.

The basic thrust of the scenario presented in Refs. Dienes:2002bg (); Dienes:2004rt () is to develop a self-consistent understanding of gauge coupling unification in the presence of large extra spacetime dimensions. For simplicity, let us imagine a -dimensional theory exhibiting a grand-unified symmetry . Let us furthermore imagine breaking this symmetry down to the Standard-Model (SM) gauge group through an orbifold compactification of the extra dimensions. For simplicity, we shall imagine that each of these extra dimensions is compactified on a circle with radius , along with an overall orbifold twist which is designed not only to preserve the zero modes of those gauge fields which survive the GUT symmetry breaking (such as the gauge bosons of our SM gauge group), but also to project out the zero modes of those remaining gauge fields (such as the and gauge bosons) which are exotic from the point of view of the SM gauge group but which were otherwise needed in order to fill out . Thus, at low energies, our spectrum consists of only the SM zero modes, and the original GUT symmetry appears broken. Indeed, the lowest-lying exotic states are the and gauge bosons which do not appear in the resulting spectrum until the first excited KK level, with masses . Of course, the full grand unification does not occur until the low-energy gauge couplings actually unify at some much higher scale . Thus, we immediately observe a remarkable feature of GUT breaking by orbifolds (as opposed to, say, the more traditional GUT breaking via a Higgs mechanism): although the actual grand unification (as evidenced through the unification of gauge couplings) only occurs at , the first experimental signatures (or “precursors”) of the impending unification are the and gauge boson states which first appear at  — a scale which is parametrically distinct from . The question then arises as to how large a separation of scales can be tolerated between the precursor scale and the unification scale . In other words, how large a compactification volume can be tolerated in such a scenario? What is the maximum allowed value of ?

This is the question addressed in Refs. Dienes:2002bg (); Dienes:2004rt (). Remarkably, what was found is that can actually grow arbitrarily large. The criteria leading to this possibility can be understood as follows. In the presence of extra spacetime dimensions of radius , the low-energy gauge couplings (as measured, say, at ) evolve upwards in energy (to an arbitrary high scale ) according to the approximate one-loop RGE’s TV (); DDG1 (); DDG2 (); DDG3 ()

(9)

where are the beta-function coefficients of the zero-mode fields, where are the beta-function coefficients associated with the field content at each excited KK level, and where is the volume of the unit ball in dimensions. It is the presence of KK states running in the loops that causes the evolution to follow a power-law behavior. As we shall shortly see, this generic form is also borne out by explicit string results within the particular six-dimensional configuration discussed at the end of Sect. II.

In a scenario with arbitrary values of , each low-energy gauge coupling experiences an independent power-law evolution and the measured low-energy couplings are grossly inconsistent with unification. However, there do exist values of for which our low-energy gauge couplings experience not only power-law evolution but also unification DDG1 (); DDG2 (); DDG3 (). One example is an accelerated, power-law evolution which usually occurs soon after the onset of the KK modes, leading to values of which are tightly constrained and often smaller than a single order of magnitude.

There is, however, a second option orbifoldguts (): all can be equal, with for all . In this case each gauge coupling continues to experience a power-law running, but the differences between the gauge couplings evolve only logarithmically. Indeed, for appropriate values of , we can reproduce a logarithmic unification which inevitably occurs at the traditional high scale . Of course, since each individual coupling experiences a power-law evolution over this entire energy range, we must be sure that none of these couplings hits a Landau pole en route to unification or otherwise accrues a value which would invalidate our overall implicit perturbativity assumptions. This generally requires that . This in turn ensures that our measured individual gauge couplings at low energies flow to extremely small (rather than extremely large) values in the UV, ultimately yielding a unified gauge coupling . Thus, in this manner, the very small values can naturally be reconciled with the measured values of the gauge couplings at low energies, all while preserving a traditional logarithmic unification of gauge couplings and a correspondingly large value for . Hence gauge coupling unification survives, even with large-volume compactifications.

There is also another way to understand this result and to verify its perturbativity. In theories such as this for which there are many degrees of freedom, an effective measure for the strength of gauge interactions is not the gauge coupling but rather the ’t Hooft coupling , where is a measure of the number of degrees of freedom running in the loops. Indeed, for any energy scale , we may take as the number of KK levels that have already been crossed, i.e., . According to Eq. (9), the individual gauge couplings all scale in the UV (i.e., for ) as . Thus the corresponding ’t Hooft couplings scale as . In other words, as originally noted in Ref. agashe (), the effective ’t Hooft couplings become independent of as increases and actually approach a UV fixed point . Moreover, this UV fixed point is perturbative so long as , or — indeed, the ’t Hooft coupling can then be interpreted as the dimensionless coupling associated with the -dimensional theory that emerges in the infinite-volume limit. Consequently, if is sufficiently large and negative, there is no obstruction to having an arbitrarily large compactification volume with . This is the underlying reason why this scenario can tolerate a large separation of scales between the GUT precursor scale and the unification scale .

It is not difficult to realize such theories in a natural way. For example, let us imagine, as in Ref. Dienes:2002bg (), that our zero-mode fields exhibit  SUSY and are those of the MSSM, while our unified gauge group is . Let us further imagine that only one extra dimension is compactified, i.e., . It then follows that the states at each excited KK level are  SUSY vector multiplets transforming in the adjoint of , with for all . This then leads to a unified perturbative fixed-point coupling .

The presence of multiplets at each excited KK level is an extremely beneficial outcome, since the presence of  SUSY in the bulk ensures that any higher-loop power-law effects are suppressed by a factor of relative to the one-loop effects. Such higher-loop effects therefore become increasingly insignificant for  DDG1 (); DDG2 (); Kakushadze:1999bb (); Dienes:2002bg (); Dienes:2004rt (). Likewise, there can be other effects (such as non-universal logarithms or contributions from brane-kinetic terms Dienes:2002bg ()) which, at first glance, also appear to have the power to eliminate the logarithmic unification in this scenario. However it can be shown Dienes:2002bg () that such effects are ultimately subleading and generally leave the unification intact.

Thus far, we have shown how the measured low-energy gauge couplings can, through power-law running associated with a large compactification volume, lead to a logarithmic gauge coupling unification at a relatively high scale . As we have seen, the principal required ingredients are the existence of complete GUT multiplets at each excited KK level, the presence of SUSY at each excited KK level, and a field content at each excited KK level such that . These properties for the excited KK states ensure that the differences between the low-energy couplings evolve at most logarithmically, that each individual coupling becomes extremely weak at the GUT scale for , and that the contributions from higher loops do not disturb our one-loop results. Properly choosing the field content of the zero modes then ensures that these couplings actually unify, just as they would have in four dimensions.

Given this field-theoretic scenario, the final step is to embed this scenario within a UV-complete theory such as string theory. However, this is not difficult to arrange: we simply identify with the four-dimensional gauge coupling in Eq. (8). Likewise, we identify with . Note that our identification of with is not meant to be a precise one, for there can be many effects which could explain a discrepancy between and . Such effects are reviewed, for example, in Ref. Dienespath (). Likewise, at first glance it may seem strange to match a one-loop “bottom-up” coupling such as with a tree-level “top-down” string coupling such as . However, this lopsided matching between a one-loop coupling and a tree-level coupling arises only because our determination of was itself the result of a tree-level string analysis. Indeed, we could equally well have performed a more complete one-loop string analysis, carefully integrating out all heavy string states before applying our matching conditions. In such a case, we would then understand the volume dependence as arising from string threshold corrections. As an example, in the toroidal 6D case (such as in the explicit example we shall present later), the result is expressed in terms of the usual moduli for the compactification, namely and . The gauge couplings are then found to behave universally as

(10)

where Angelantonj:2015nfa (); Abel:2016hgy ()

(11)

Here and with , where is the compactification two-volume. In Eq. (11) we see both the leading terms and the logarithmic contribution from the running of the sector between the KK scale and the string scale.

Given that we are forced to embed the GUT-precursor structure into string theory, one natural issue is to determine the scale at which gravitational effects become strong. Since and , it follows that . At first glance, this might seem to imply that gravitational effects do not arise until far beyond the string scale. However this scenario involves a large volume of compactification, and it is well known that under such circumstances the actual quantum-gravity scale is given by where is the volume of compactification. We thus consistently find that , implying that the effective quantum-gravity scale is not greatly separated from the string scale, just as occurs in more traditional scenarios involving Planck-scale compactification volumes.

Combining the above relations, we find that

(12)

Thus the Planck scale , the string scale , and the Kaluza-Klein scale are all balanced together in any self-consistent heterotic string-theoretic scenario. It is useful to examine some representative cases. If and we identify , we find . For , this implies  GeV, while for this implies  GeV. Indeed, taking larger values of only increases the value of . From the perspective of the low-energy theory, this is an extremely large scale for SUSY-breaking. We nevertheless find from Eq. (3) that for and for , assuming that in each case. Thus the dangerous one-loop dilaton tadpole is extremely suppressed and essentially zero for all practical purposes.

It is important to note that the overall scaling relations we have been working with are ultimately governed by , the universal beta-function coefficient associated with the matter content of the excited KK states. By contrast, the value of the unification scale is set by the values of the individual beta-function coefficients associated with the zero-mode states. Thus, there remains the freedom — just as in all field-theoretic GUT scenarios — to choose our low-lying matter content in such a way as to alter these beta-function coefficients and thereby adjust the unification scale. In this way, it might even be possible to bring significantly below the traditional unification scale. Continuing to identify with would then lead to a self-consistent scenario in which  GeV, with correspondingly reduced even further, perhaps even all the way into the TeV range. Indeed, taking  TeV within Eq. (12), we find that  GeV for , whereupon we see that . Even for we find  GeV, whereupon . Thus, even in such cases with significantly reduced string scales, we continue to find that the one-loop dilaton tadpole is extremely suppressed. Such scenarios thus retain dilaton stability and incorporate not only an effective TeV-scale breaking of SUSY but also GUT-precursor states that are potentially observable at the TeV scale — orders of magnitude lower than the scale of gauge coupling unification! Such states would have the gauge quantum numbers of leptoquarks and would thus give rise to dramatic collider signatures.

iii.3 The structure and ubiquity of -Susy

As we have seen, non-supersymmetric strings can be made stable in a consistent fashion if the excited KK modes consist of GUT representations falling into supermultiplets. Therefore, in order to obtain the Standard Model for the zero modes, our compactification must not only break the GUT gauge symmetry but also simultaneously break the remaining supersymmetry. As we now discuss, this inevitably gives rise in the resulting theory to a structure involving what we call “entwined SUSY” (-SUSY). We begin by describing more explicitly what we mean by entwined-SUSY, both in terms of the allowed spectrum as well as the allowed couplings. We shall then discuss why this structure arises.

As an example, let us consider a theory in which an underlying GUT model contains two generations of chiral supermultiplets, and . Here the subscripts indicate the charges under a horizontal symmetry which we will refer to generically as . (These multiplets may also carry other horizontal charges, but only one horizontal charge is needed in order to illustrate the entwined-supersymmetric structure.) Under the group decomposition , we recall that . Our two chiral multiplets and thus have states consisting of

(13)

where the hypercharges (which we do not show) are the canonical ones. Indeed, this is the matter content that would emerge upon compactification without the crucial Scherk-Schwarz twists.

Implementing the twists then eliminates part of this matter content. Of course, which states survive and which are projected out depends on the details of the relevant Scherk-Schwarz twists and GSO projections. In many simple string constructions, these projections would eliminate either one or the other of these supermultiplets. Likewise, supersymmetry would be broken if the internal structure of each multiplet was also destroyed, leaving behind bosonic and fermionic states that could no longer be paired with each other. However, what we find in the configuration described above — where the last stage of compactification breaks the supersymmetry and GUT symmetry simultaneously — is that the projections instead lift the masses of certain states according to a non-trivial combination of their SM representations, horizontal charges and spin-statistics. Indeed, what remains at the massless level are a set of states which together fill out a single light “fake” supermultiplet which we shall denote :

(14)

All other components from the original pair of ’s in Eq. (13) are given masses of order the compactification scale.

It is immediately apparent from the structure of the multiplet in Eq. (14) that the SUSY-breaking is “entwined” with the horizontal charges in a non-trivial way. Indeed, the matter spectrum is not symmetric under a supersymmetry transformation alone, but only a supersymmetry transformation coupled with a permutation of charges. Thus, the supersymmetry is completely broken in the resulting theory. For example, no massless gravitinos or gauginos survive in the massless spectrum. Nevertheless, due to the controlled structure of the SUSY-breaking, a residual imprint of the original supersymmetry remains. This is our “entwined” SUSY (e-SUSY).

A similar entwining also occurs for the other GUT multiplets. For example, suppose that (e.g., as dictated by anomaly cancellation) the content of our original GUT model fills out entire SM generations with the inclusion of a representation and a representation. In the spontaneously broken theory, the entwined 10 multiplet is then given by

(15)

where we have adopted the same convention as for the ’s, namely that goes with the doublet fermions and singlet bosons, and vice-versa for . Note that anomaly cancellation (such as cancellation of the anomalies) requires the existence of a second -multiplet which we do not show with the horizontal charges negated. This will be present in the explicit example to be presented in Sect. IV.

For vector-like pairs, in particular the Higgses, anomaly cancellation is achieved with a slightly different entwining in which the supersymmetry transformation is coupled with a permutation of with fundamental. Let us assume that there is a vector-like pair of Higgs supermultiplets and . In contrast with the matter ’plets, entwined SUSY leaves light the scalar doublets and as well as a vector-like pair of fermionic color triplets. The entwined multiplet takes the form

(16)

where and are color triplets. Of course the -SUSY structure is a feature of the matter and Higgs sectors only. In particular, it does not extend to the gauge sector; indeed, the gauginos are heavy, as we have said, and there are no partners for the gauge bosons.

Entwined SUSY is not just a property of the spectrum — it also governs the allowed couplings. As an example, let us consider the part of the superpotential of the original GUT theory that encapsulates the down and lepton Yukawa couplings:

(17)

Note that both terms share the same coupling . We have also assumed that the Higgs descends from a higher-dimensional gauge boson and thus is an off-diagonal component of the adjoint representation of a larger, broken symmetry. Indeed, this is always the case if the Higgs is a state from the Neveu-Schwarz Neveu-Schwarz (NS-NS) sector; as we shall shortly see, such states are generic.

We can divide the superpotential in Eq. (17) into two components, , where involves those matter supermultiplets whose fermions remain light after SUSY breaking while involves the supermultiplets whose bosons remain light. Keeping only those pieces that include the Higgs doublets, we then have

(18)

where denote the complete supermultiplets. However, the crucial point is that the scalar Higgs in the light theory couples to an entire -multiplet, i.e., both to fermions and to their -partners, with degenerate couplings. Thus, for example, the quadratically divergent contributions to the Higgs squared-mass from these multiplets still cancels, much as in genuine SUSY.  Indeed, this -SUSY structure occurs for every pair of couplings in the original GUT theory that are already independently invariant under a permutation of . For example, if only the first piece of the superpotential in Eq. (17) had existed, then only the first pieces of and would have been present, and -SUSY would have been broken in these couplings. A typical theory contains examples of both kinds of coupling. In this connection, we remark that such a structure also has the potential to solve the Higgs hierarchy problem. We shall comment on this below.

At first sight, the emergence of e-SUSY may seem surprising. However, in the present context, this structure is essentially forced upon us. To see why, we begin by recalling that in our scenario, we are compactifying from ten dimensions to four dimensions in two stages: first , and subsequently . Moreover, the existence of the GUT-precursor structure means that only the second stage of the compactification may break the GUT symmetry. Indeed, if this were not the case, then there would be mass splittings between components of a single GUT multiplet. However, there are two components to this final stage of compactification. The first is an action (i.e., a set of phases) on the fields associated with the Scherk-Schwarz twist. By itself, this would break the 4D theory from  SUSY to  SUSY, resulting in a non-chiral theory. By contrast, the second ingredient is the aforementioned orbifold action which, by itself, would break  SUSY to  SUSY.  In principle, either of these is a suitable place in which to embed the breaking of the GUT symmetry, and indeed both options lead to mass splittings between different components of a single GUT multiplet. In this paper, we will without loss of generality assume that the GUT symmetry is broken via the latter procedure.

Let us now consider the properties of the resulting spectrum. Relative to the orbifold action of , some sectors of the theory will be untwisted and some will be twisted. Of course, the twisted sectors remain fully supersymmetric because they are blind to the large radius. Thus we immediately see that it is only the untwisted sectors which exhibit the SUSY-breaking and the eventual entwined SUSY.  As discussed above, the GUT-precursor structure indicates that any SUSY- and GUT-breaking that occurs in these sectors must be driven entirely by non-trivial GSO phases — i.e., by a non-trivial Scherk-Schwarz action. The effect of these phases is to lift the masses of certain components of the SUSY GUT multiplets to , where is the typical compactification radius.

We can gain a simple understanding of which components will have their masses lifted as follows. In general, each string state has a corresponding charge vector which we may write in the form

(19)

Here and denote the charges of this state under the Standard-Model and gauge symmetries, while generally denotes charges under other gauge symmetries which may be viewed as horizontal relative to those of the Standard Model. Likewise, are charges indicating the spacetime helicity (spin-statistics) of the state, while denote its internal -charges.

Note that these charge vectors have natural identifications within the special case of the heterotic string. For heterotic strings in six dimensions, the charge vectors of the string states fill out a -dimensional Lorentz self-dual lattice where the 20 dimensions correspond to the bosonic (or gauge) side of the heterotic string and the 8 dimensions correspond to the superstring (or spacetime) side. For SM-like strings, we may in general identify three dimensions within the 20 as corresponding to charges, while we may identify two others as corresponding to . Relative to these gauge groups, we may regard the remaining 15 dimensions as corresponding to horizontal symmetries. These three sets of charges therefore correspond to , , and . Likewise, on the spacetime side, two lattice dimensions correspond to the spacetime helicities (spin-statistics). These are therefore the charges we have denoted . Note that the remaining six dimensions on this side are purely internal, and their charges may be viewed as -charges. String consistency constraints concerning the worldsheet supercurrent correlate these charges with , and thus the -charges, like , are sensitive to whether a given state is bosonic or fermionic in spacetime.

In general, different states within a given GUT multiplet will have different charge vectors. For example, a complete supermultiplet in the GUT theory might decompose into fermionic and bosonic pieces with charge vectors

(20)

where we have left unspecified in order to allow for different Standard-Model charges. In an untwisted sector the bosons typically have , which we have adopted above for concreteness.

The question is therefore to determine which of the massless states in such multiplets will have their masses lifted by the Scherk-Scherk twist. In general, for a given field and a compactified direction with coordinate and radius , such a twist takes the form

(21)

where the vector specifies the particular twist and takes the form

(22)

and where the dot-product is Lorentzian (i.e., gauge minus spacetime). In general, implementing this twist raises the masses of those states which carry a net Scherk-Schwarz charge. More specifically, in the presence of a universal compactification radius , we see from Eq. (21) that any state with a charge vector will experience a shift in its KK mode number of the form . This implies that the mass of a previously massless state with charge vector now becomes

(23)

Of course, if is an integer, then the KK mode numbers of our states merely shift by an integer. There is thus always another KK mode which now becomes massless and which takes the place of the original state in the sense that it has the same spacetime properties. This then explains the restriction “mod (1)” in Eq. (23). We conclude that only those states for which survive in the massless spectrum.

Given this, each choice of -vector corresponds a specific resulting pattern of SUSY-breaking and GUT-breaking. Certain aspects of the required twist are then obvious. First, the Scherk-Schwarz twist has to distinguish bosonic states from fermionic states. In principle, this can be accomplished by having this twist be sensitive to either or ; for technical reasons the choice is more natural. Likewise, in order to break the GUT group, the twist must be sensitive to or . Finally, string self-consistency conditions then require that the twist also generically act on the different horizontal charges of the states .

As an example, let us suppose that we wish the Scherk-Schwarz twist to act on (in order to distinguish bosons from fermions), and to simultaneously break the GUT symmetry by acting on and but not on . A relevant Scherk-Schwarz action for the projection above is then given by the vector

(24)

The states from our original supermultiplet that remain light therefore obey

(25)

where we define . Clearly there is some model-building freedom in the choice of and the corresponding distribution of charges among the matter multiplets.

In order to determine the resulting massless spectrum, it is necessary to discuss the values of the trace charge , which depends on how the representations are constructed. In a typical construction the matter and come from spinor representations of a larger gauge group [e.g., ], and in this case the matter doublets and have while and have and and have . Meanwhile, given our previous assumptions, the Higgses appear in the bifundamental (with one factor in the group and the other factor in a hidden gauge group). They therefore have .

Adopting these charges and applying the projection in Eqs. (25) with the charges in Eq. (III.3), we see that the remaining light matter fermions are (and Higgs triplets). Likewise, the light scalars are . Thus, the massless left-handed fermion matter doublets have after applying the Scherk-Schwarz mechanism, while their bosonic “pseudo-superpartners” actually come from the states with . Conversely the massless right-handed matter fermions have and their bosonic “pseudo-superpartners” have . All other states acquire a mass . Finally, the original GUT theory must be free of anomalies, which requires that two -twisted generations descend from four GUT generations (with corresponding horizontal charges ).

This is precisely the e-SUSY structure described above. The specific distribution of charges may differ from theory to theory, but the general robust feature is that differently-charged components comprise -supermultiplets. Indeed, the particular distribution of charges described above is quite typical. The crucial feature of the Scherk-Schwarz mechanism that results in this structure is that the breaking of supersymmetry and gauge group occur simultaneously in the underlying string construction. This leads to a correlation between and the -charges, and hence between and the spacetime spins of the components of the -supermultiplets that actually descend from different supermultiplets of the original SUSY GUT.

Note that this mechanism operates only for supermultiplets that carry horizontal charges overlapping with the Scherk-Schwarz action .  S