Contents

IFT-UAM/CSIC-12-55

The Intermediate Scale MSSM, the Higgs Mass and F-theory Unification

[3mm]

L.E. Ibáñez, F. Marchesano, D. Regalado and I. Valenzuela

[3mm] Departamento de Física Teórica and Instituto de Física Teórica UAM-CSIC,

[-0.3em] Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

[6mm] Abstract

[7mm]

Even if SUSY is not present at the Electro-Weak scale, string theory suggests its presence at some scale below the string scale to guarantee the absence of tachyons. We explore the possible value of consistent with gauge coupling unification and known sources of SUSY breaking in string theory. Within F-theory unification these two requirements fix GeV at an intermediate scale and a unification scale GeV. As a direct consequence one also predicts the vanishing of the quartic Higgs SM self-coupling at GeV. This is tantalizingly consistent with recent LHC hints of a Higgs mass in the region 124-126 GeV. With such a low unification scale GeV one may worry about too fast proton decay via dimension 6 operators. However in the F-theory GUT context is broken to the SM via hypercharge flux. We show that this hypercharge flux deforms the SM fermion wave functions leading to a suppression, avoiding in this way the strong experimental proton decay constraints. In these constructions there is generically an axion with a scale of size GeV which could solve the strong CP problem and provide for the observed dark matter. The price to pay for these attractive features is to assume that the hierarchy problem is solved due to anthropic selection in a string landscape.




1 Introduction

The LHC is already providing us with very important information on the physics underlying the Standard Model (SM) symmetry breaking process. A first piece of information are the constraints on the mass of the Higgs particle which is either heavier than 600 GeV or else confined to a region in the area GeV. In fact after the 7 TeV run there are important hints suggesting a Higgs mass in the region GeV from both CMS and ATLAS [1, 2]. A second piece of information is the absence of any trace of physics beyond the Standard Model (BSM). In particular there is at present no sign of squark and gluinos below 1 TeV, at least if they decay via the standard R-parity conserving channels in the Minimal Supersymmetric Standard Model (MSSM).

A Higgs mass around GeV is in principle good news for supersymmetry. Indeed such a value is consistent with the MSSM which predicts a mass GeV for its lightest Higgs scalar. On the other hand getting such a Higgs mass within the MSSM typically requires a very massive SUSY spectrum with e.g. squarks at least in the TeV region [3]. This massive spectrum requires in turn a fine-tuning of the parameters at the per-mil level. If after the run at 14 TeV the LHC sees no sign of supersymmetry or any other new physics BSM, the required fine-tuning will increase further and we will have to consider seriously the possibility that indeed the Electro-Weak (EW) scale is fine-tuned and selected on anthropic grounds [4, 5, 6, 7].

If the EW scale is fine-tuned and low-energy SUSY does not play a role in the hierarchy issue, one may think of resurrecting good old non-SUSY unified theories like . We have to recall however the limitations of non-SUSY unification. Unification of gauge coupling constants, which works so well with the MSSM, fails in the non-SUSY case. Furthermore in minimal models the unification scale is around GeV and the proton decays too fast via dimension 6 operators. We also loose the existence of a natural candidate for dark matter to replace the neutralinos in the MSSM.

There is however another hint telling us that a non-SUSY desert up to a unified scale and a fine-tuning of the Higgs mass is unlikely. Indeed if a SM Higgs mass is around GeV one knows what is the value of the Higgs quartic coupling at the EW scale. One can then extrapolate its value up in energies using the Renormalization Group Equations (RGE). Due to the large top quark Yukawa coupling decreases at higher energies and in fact seems to vanish at a scale around GeV, with the precise scale depending on the precise value of the top-quark mass and the strong coupling constant [8, 9, 10, 11, 12, 13, 14, 15, 16]. If this is the case the theory becomes metastable before reaching the unification scale.

In any event, supersymmetric or not, one expects any unified theory to be combined with gravitation into an ultraviolet complete theory. At present the best candidate for such completion is provided by string theory. It is thus natural to try to address the unification issue within the context of string theory. In the last few years an interesting embedding of the unification idea into F-theory has been the subject of much work (for reviews see [17, 18]). These so called F-theory GUTs have some similarities with standard field theory models but differ in some important aspects. Thus, e.g., the breaking of down to the SM is produced by the presence of hypercharge fluxes in the compact dimensions instead of an explicit Higgs mechanism. As we will see this leads to several physical effects on both the gauge and Yukawa couplings which modify several aspects of field theory GUTs.

In the present paper we address the embedding of the SM or its SUSY version into the scheme of F-theory unification without any prejudice about the size of the SUSY breaking scale . At this scale it is assumed that soft terms break the SUSY SM 111The SUSY spectrum could be that of the MSSM of some extension with e.g. extra Higgs doublets or triplets, see below. Whenever we write MSSM we also mean this kind of extensions, which will require minimal modifications. into the minimal SM. We study what the scale of unification and SUSY breaking should be in order to obtain 1) correct gauge coupling unification, 2) sufficiently suppressed proton decay and 3) consistency with a SM Higgs in the GeV region.

F-theory unification has specific hypercharge flux threshold corrections [19, 20, 21] to the inverse couplings which are proportional to the inverse string constant . At strong coupling they are suppressed and correct gauge coupling unification is obtained with the MSSM spectrum and TeV in the usual way. On the other hand, if one wants to remain at weak coupling, the threshold corrections are too large and spoil MSSM unification (unless extra effects from particles beyond the MSSM are included). However leaving the SUSY breaking scale free with TeV one finds that the threshold corrections have the required form and size to yield correct gauge coupling unification without the addition of any extra particle beyond the MSSM content. We argue that there are three independent arguments suggesting is at an intermediate scale GeV with gauge unification then fixing a unification scale GeV. First, SUSY breaking induced by closed string fluxes gives rise generically to soft terms of order GeV. Second, if indeed the SM Higgs self-coupling vanishes at a scale of order GeV, as seems to be implied by a GeV Higgs, this may be an indication of a SUSY boundary condition with tan. We show that this boundary condition is quite generic in string constructions in which a Higgs field is fine-tuned to be massless. Finally, in F-theory GUT’s there is a natural candidate for a string axion with decay constant , which is in the right region GeV for axionic dark matter if GeV.

In this ISSB222Intermediate Scale Supersymmetry Breaking. framework the unification scale is relatively low, GeV and one may worry about fast proton decay via dimension 6 operators. Again, the fact that the breaking of the symmetry down to the SM proceeds due to a hypercharge flux background rather than a Higgs mechanism modifies the expectations. Indeed, in field theory the gauge coupling of the gauge bosons to fermions remains unchanged before and after symmetry breaking. However if symmetry breaking is induced by hypercharge fluxes, the coupling to fermions may be substantially suppressed due to the fact that the profile of the corresponding wave functions is modified. We describe this novel effect in detail by using local F-theory wave-functions of matter fields.

The outline of this paper is as follows. In the next chapter we present a brief review of F-theory unification with emphasis on the particular aspects which play a role in the following sections. In chapter 3 we discuss gauge coupling unification and the scales of SUSY breaking and unification naturally arising. In chapter 4 we discuss the value of the quartic Higgs coupling at within the context of string compactifications, describing how tan comes naturally. In the following section we discuss how the fine-tuning of a massless SM Higgs can proceed and the one-loop stability of the tan boundary condition. In chapter 6 we address the issue of proton decay suppression and in chapter 7 we discuss other phenomenological consequences and in particular how an axion with an appropriate decay constant naturally appears within this framework. We briefly discuss the case of Split-SUSY in section 8 and leave section 9 for some final remarks and conclusions. Three appendices complement the main text.

2 and F-theory unification

F-theory [22] may be considered as a non-perturbative extension of Type IIB orientifold compactifications with 7-branes. This class of compactifications have two main phenomenological virtues compared to other string constructions [17, 18]). First, in Type IIB compactifications it is well understood how moduli could be fixed in the presence of closed string fluxes and non-perturbative effects. Secondly, particularly within F-theory, GUT symmetries like appear allowing for a correct structure of fermion masses (in particular a sizeable top quark mass). Here we just review a few concepts which are required for the understanding of the forthcoming sections. Our general discussion applies both to perturbative Type IIB and their F-theory extensions but we will refer to them as F-theory constructions for generality.

Figure 1: Scheme of an F-theory GUT. The six extra dimensions are compactified on whereas the degrees of freedom are localized on a 4-cycle submanifold . The gauge bosons live on the bulk of but the chiral multiplets are localized on complex matter curves. At the intersection of two matter curves with a Higgs curve a Yukawa coupling develops.

In Type IIB orientifold/F-theory unified models the symmetry arises from the worldvolume fields of five 7-branes with their extra 4 dimensions wrapping a 4-cycle inside a six dimensional compact manifold , see fig.1. The matter fields transforming in 10-plets and 5-plets have their wave functions in extra dimensions localized on complex curves, the so called matter curves. These matter curves, which have two real dimensions, may be understood as intersections of the 7-branes with extra 7-branes wrapping other 4-cycles in .

In order to get an idea of the relevant mass scales one can use results from perturbative Type IIB orientifolds. The string scale is related to the Planck scale by [17]

(2.1)

where is the volume of the 6-manifold and in the string coupling constant. is the inverse string tension. Note that one can lower by having a large volume (or decreasing ), so that the string scale is in principle a free parameter.

Now, the volume of the 4-fold which is wrapped by the 7-branes is independent from the overall volume of . This volume however is related to the inverse GUT coupling constant . In particular one has at tree level

(2.2)

with the gauge kinetic function. Parametrizing one then has

(2.3)

where we define . This is slightly below the string scale (i.e. for and one has ). This scale can be identified with the GUT scale at which is broken down to the SM. Indeed in F-theory GUTs there are no adjoint Higgs multiplets nor discrete Wilson lines available and it is a hypercharge flux background which does the job [23, 24]. These fluxes go through holomorphic curves inside and they are quantized, integer. Thus on dimensional grounds one has and indeed one can identify with the GUT scale.333Hypercharge fluxes have an additional use in F-theory GUT’s. Indeed, by appropriately choosing these open string fluxes one can get doublet-triplet splitting in the Higgs 5-plet, see refs. [23, 24, 18, 17] for details. However, as we will remark later, doublet-triplet splitting is not strictly necessary in our scheme, and so the constraints that are usually required on the hypercharge flux in order to achieve doublet-triplet splitting can be relaxed in our setup.

We want to consider here the case in which slightly below the unification scale we have unbroken SUSY with an MSSM spectrum (or some slight generalization, see below). One reason for that assumption is that such class of vacua have no tachyons which could cause any premature instability in the theory. We will find additional a posteriori justifications for such an option in the forthcoming chapters. We will however allow for SUSY to be broken in the MSSM sector at a scale to be determined. It is however important to realize that there is a natural scale of SUSY breaking in Type IIB/F-theory compactifications.

Indeed, a most natural source of SUSY breaking is the presence of closed string fluxes in such vacua. More precisely, it is well known that e.g. generic RR and NS 3-form fluxes in Type IIB orientifolds induce SUSY-breaking soft terms [25]. These are also the fluxes which play a prominent role in the fixing of the moduli in these vacua. Since these fluxes live in the full CY and are quantized on 3-cycles, the said soft terms scale like . One thus finds for the size of soft terms [17]

(2.4)

with c some fudge factor. Taking and taking into account eq.(2.3) one thus has an estimation for 444Note that setting GeV one would obtain a scheme with soft terms around 1 TeV, which would be consistent with a SUSY solution of the hierarchy problem. However that would require also a string scale of order GeV and MSSM gauge coupling unification would be lost. Alternatively one can set GeV consistent with MSSM gauge coupling unification if the effect of fluxes is somehow suppressed. Possible ways to suppress it would be assuming some fine-tuning in the flux or some warp factor leading to a flux dilution. This is the implicit assumption in models with flux induced SUSY breaking, GeV and a standard SUSY solution to the hierarchy problem.

(2.5)

We will discuss other possible sources of SUSY breaking in section 4.

Although it will not play a relevant role in our discussion, let us briefly mention how the Yukawa couplings appear in the framework of F-theory unification. As we said the matter multiplets of the MSSM are confined in complex matter curves within the 4-fold . Yukawa couplings appear at triple intersection points in in which two matter curves involving 10-plets and 5-plets cross with a matter curve containing the Higgs 5-plets, see fig.1. The Yukawa couplings may be computed as in standard Kaluza-Klein compactifications from triple overlap integrals of the form

(2.6)

where are family indices. The wave functions have a Gaussian profile so that one only needs local information about these wave functions around the intersection points in order to compute the Yukawa couplings. This local information may be extracted from the local equations of motion which may be solved so that quite explicit expressions for these wave functions may be obtained. We will use these local wave functions when discussing the proton decay suppression in section 6.

As a summary we see that F-theory unification allows for a general structure of scales . In what follows we will discuss how constraints from gauge coupling unification and the Higgs mass fix these scales.

3 Gauge coupling unification and hypercharge flux

In order to check for gauge coupling unification we will assume that at some scale the symmetry is broken by hypercharge fluxes down to the SM group. We will asume that after this breaking the particle content is that of the MSSM (although we will allow for some variation below). However, unlike in the usual low energy SUSY scenario, we allow the scale of SUSY breaking on the MSSM to be a free parameter. We know that the standard MSSM prediction for gauge coupling unification [26] with TeV is quite successful. On the other hand for we have the SM below and we know that coupling unification fails. On the basis of this one could conclude that gauge coupling unification forces to be close to the weak scale. Interestingly enough, the breaking of the symmetry via fluxes has a novel type of threshold corrections [19, 20, 21] compared to the field theory case, as we now describe.555These corrections result from the expansion in powers of fluxes of the Dirac-Born-Infeld plus Chern-Simmons (DBI+CS) action of the 7-branes, see e.g.[17] for a review. To leading order the gauge kinetic function for the group within the 7-branes is given by the local Kähler modulus whose real part is proportional to , consistently with eq.(2.2). However in the presence of hypercharge fluxes the gauge kinetic functions get corrections [20]

(3.1)

where is the complex dilaton and are fluxes along the U(1) contained in the U(5) gauge group of the 7-branes which are needed for technical reasons but are not relevant in our discussion. It turns out that in order to get rid of exotic matter massless fields beyond those of the MSSM the topological condition should be fulfilled [23, 24]. This implies that at the compactification scale one has the condition

(3.2)

which is a generalization of the standard relationship . In addition one also obtains

(3.3)

Thus the size of the threshold corrections is determined by the inverse of the string coupling . The corrections by themselves would imply an ordering of the size of the fine structure constants at given by

(3.4)

We will be neglecting in what follows other possible sources of threshold corrections like the loop contributions of KK massive modes. The corrections in eq.(3) may in fact spoil the standard joining of gauge coupling constants in the MSSM, which works quite well, unless they are suppressed by assuming . Furthermore the above ordering of couplings goes in the wrong direction if one wanted to use such corrections to further improve the agreement with experiment [20].

Interestingly enough, in our setting with undetermined the corrections have just the required form and size to get consistency with gauge coupling unification without the addition of any extra matter field beyond the MSSM (see also ref.[27]). The one-loop renormalization group equations lead to the standard formulae

(3.5)

where are the one-loop beta-function coefficients of the SM and the MSSM respectively. These are given by and . Combining these equations and including the boundary condition (3.2) (which amounts to allowing for the above threshold corrections) one obtains

(3.6)

In our case this yields

(3.7)
Figure 2: Constraints on and from gauge coupling unification (black line) and closed string flux induced SUSY breaking (red line). The vertical (blue) band shows the region of at which the SM Higgs coupling vanishes for a Higgs mass in the range GeV and GeV, as extracted from ref.[9, 11]. The horizontal line shows the value of the axion decay constant for the selected unification mass .

This is displayed by the black line in figure 2. We have used the central values of the couplings

(3.8)
(3.9)
(3.10)

One observes that one can get consistent unification for values of up to slightly below GeV, which is required by the condition . The unification scale has also a lower bound at the same scale. If however we impose that SUSY breaking is induced by closed string fluxes with as explained in the previous section, the values of both and are fixed yielding

(3.11)

Thus one gets correct coupling unification consistent with closed string flux SUSY breaking for SUSY broken at intermediate scale GeV and a slightly low unification scale of order GeV. This immediately poses an apparent problem with proton decay that we will deal with in section 6.

It is also interesting to display the value of as a function of from eq.(3.3). This is shown in fig. 3. For the values in eq.(3.11) one finds . This shows that the string coupling here is in a perturbative regime. On the other hand for values TeV, corresponding to standard MSSM low-energy supersymmetry one needs . Note that in the context of F-theory the dilaton value varies over different locations in extra dimensions and may be large or small, so both situations are possible in F-theory GUTs.

Another interesting point is that the unification line in fig. 2 does not change if in the region there are incomplete 5-plets, as equation (3.7) does not change. Thus for example the curve remains the same if the partner of the Higgs doublets, the triplets transforming like , remain in the spectrum below . These triplets are potentially dangerous since their exchange give rise to dimension 6 proton decay operators. The rate is above experimental limits unless GeV [28], see section 6. That is why in GUTs one needs to perform some form of doublet-triplet splitting so that the Higgs fields remain light but the triplets are superheavy. In our case however these triplets will get a mass of order GeV anyhow so they may be tolerated below and no doublet-triplet splitting is necessary. The presence of these triplets does however affect the size of the threshold corrections and . In this case one gets typically smaller which slowly grows as increases, see fig. 3. For GeV one gets .

Figure 3: The string dilaton coupling constant versus for consistent gauge coupling unification. Left: With an MSSM content in the region ; Right: With an additional vector-like triplet set in that region.

4 The quartic coupling and the Higgs mass

We have seen in the previous section how this ISSB framework is consistent both with gauge coupling unification and flux-induced SUSY breaking. Interestingly enough it has been recently found that if a non-SUSY SM Higgs is around 125 GeV, the SM RGE of the quartic self-coupling seems to drive it to a vanishing value at around GeV or so (see e.g. [10, 9]).

The question is whether there is any SUSY/string based scheme in which that happens naturally. Note that the Intermediate Scale SUSY Breaking (ISSB) described above corresponds to a variant of the High Scale SUSY Breaking (HSSB) scheme of Hall and Nomura in ref.[6]. This is just assuming a MSSM structure above a very large SUSY scale . All SUSY partners are heavy but there is still some imprint left of the High Scale SUSY in the Higgs sector. Indeed out of the two scalars in the MSSM only one linear combination remains light below , i.e.

(4.1)

Then there is a quartic self-coupling with [5, 6]

(4.2)

which is inherited from the D-term scalar potential of the MSSM. As we said it has been shown [9, 10, 11, 12, 13, 14, 15, 16] that, starting at low-energies with a SM Higgs with a mass around 124-126 GeV and running up the SM self-coupling up in energies this coupling tends to zero around a scale GeV (see fig. 4). This would be consistent with the above High Scale SUSY Breaking scheme if at the scale one had tan , so that .

An interesting question is thus under what conditions one naturally gets tan . The general form of Higgs masses in the MSSM is666If in addition to the Higgs doublets there remain Higgs triplets below , similar mass matrices will appear for them. However there will be no anthropic selection of light scalar triplets. So doublet-triplet splitting would be purely anthropic.

(4.3)

where we will take real. The condition for a massless eigenvalue is . The massless eigenvector is then

(4.4)

with sin . So in order to have a massless Higgs with tan one needs to have the conditions

(4.5)
(4.6)

We will take the negative sign in (4.4) from now on. The first condition points to an underlying symmetry under the exchange of and , possibly slightly broken. The second condition does not necessarily imply any underlying symmetry, it is rather a fine-tuning constraint which has to be there anyhow if we want to get a light Higgs. So this could be selected on anthropic grounds.

One can consider as a first option the direct construction of non-SUSY compactifications. Examples of this could be e.g. theories obtained from branes sitting at non-SUSY orbifold singularities (see e.g., ref.[29]). In this class of theories the particles in the spectrum, typically involving both fermions and scalars, have no SUSY partners to start with. These are however problematic since the spectrum generically contains tachyons from the closed string sector which destabilize the theory. So we will restrict ourselves to theories in which there is an underlying SUSY which is spontaneously broken to . This will guarantee the absence of closed or open string tachyons from the start.

There are several possible sources for spontaneous SUSY breaking in the IIB/F-theory context which may arise from open or closed string fluxes 777In the F-theory context both closed and open string fluxes have a common origin as fluxes. which we discuss in turn.

i)SUSY breaking terms and open string fluxes

In Type IIB/F-theory in the large volume limit the Higgs fields will appear as KK zero modes. Open string fluxes, like the mentioned above are in general present in order to generate chirality and symmetry breaking. These fluxes may induce also Higgs masses and SUSY-breaking terms as in eq.(4.3). We now review how such mass terms may appear in Type II toroidal orientifolds as discussed in [30]. In this reference a large class of non-SUSY Type IIA orientifolds with SM group and three chiral generations is discussed in terms of D6-branes intersecting at angles. These models may be converted into Type IIB orientifolds with D7-branes by the duality that relates intersection angles between two branes into magnetic fluxes at their overlap, through the map . In these non-SUSY models (see appendix A for a short review) the Higgs fields appear from the exchange of open strings between an stack of branes and a brane or its orientifold mirror . The underlying torus is factorized as and the branes and are separated in the second torus by a distance . This means that the Higgs fields have a mass term proportional to this distance. In addition the magnetic fluxes induce non-SUSY mass contributions. One gets a structure (in the dilute flux limit) [30]

(4.7)
(4.8)

where are fluxes from ’s in the branes going through the first and third torus, see appendix A. Note that the first contributions correspond to a -term and that it automatically implies eq.(4.5). This happens because one may understand and as coming from the same hypermultiplet before SUSY is broken by the fluxes. On the other hand the size of depends on the size and alignment of fluxes. For one recovers unbroken SUSY in the Higgs subsystem. The D-term quartic selfcoupling is given by and, since this is a dimension 4 operator, it remains the same for . In principle for fixed one can fine-tune the fluxes so that eq.(4.6) is met. Thus fine tuning of fluxes (or distance ) yields a massless Higgs multiplet corresponding to tan.

Note that the open string flux misalignment corresponds to a D-term SUSY breaking. This means that by itself this can only give SUSY breaking scalar masses but no gaugino masses (Higgssinos get SUSY masses for ). Thus this class of SUSY breaking should be supplemented by further sources if below we want to get just the content of the SM.

The above structure is generic and appears in any type II configuration where one can construct D-brane sectors with an hypermultiplet or a similar spectrum. In type IIB models with intersecting D7-branes or in F-theory GUTs such sectors arise quite naturally, since at the six-dimensional intersection of two 7-branes in flat space lives a 6d hypermultiplet that is equivalent to the 4d hypermultiplet of the construction above [31]. Hence, in order to reproduce the above structure for the Higgs sector, one may consider the case where the Higgs matter curve yields a non-chiral, N=2 subsector of the theory. As the presence of a net flux over a matter curve induces a 4d chiral spectrum arising from it, the easier way to preserve the N=2 structure is to impose that the integral of any relevant flux vanishes over . Note that in supersymmetric SU(5) F-theory models this option is usually not considered, since in order to achieve doublet-triplet splitting a net hypercharge flux is required to thread the Higgs curve(s). However, as mentioned above in the present scheme we are not constrained by the amount of Higgs triplets at the scale , and one may indeed consider the case where .

In that case both and arise from the same curve , and one may easily implement the mass structure (4.3). Just like for type IIA non-SUSY models, the term arises by inducing a non-vanishing D-term on this sector, which in this case is induced by worldvolume fluxes on which do not satisfy the condition , being the Kähler form on . For instance, let be the complex coordinate of along and the one transverse to it. Then if the flux felt by the doublets in is of the form the D-term condition reads [32, 33]. Hence, the off-diagonal terms in (4.3) read and arise whenever such vanishing D-term condition is not met.

Finally, the diagonal terms of the mass matrix (4.3) will correspond to a -term. In a D7-brane setup dual to the toroidal models of appendix A this mass term appears by simply switching on a continuous or discrete Wilson line along . However, as mentioned before Wilson lines are typically not available in F-theory GUT models, and so the -term cannot be generated by this mechanism. Instead, such supersymmetric mass term can be induced by the presence of closed string fluxes (see below) or at the non-perturbative level. Indeed, a -term may appear at the non-perturbative level from string instanton effects [34] (see [35, 17] for reviews). Such -terms are automatically symmetric under exchange and hence respect the above structure. As we said, stringy instanton effects are of order exp and for could give rise to -terms of the appropriate order of magnitude .

ii) SUSY breaking terms from closed string fluxes and modulus dominance

Mass terms for scalar fields may also appear in the presence of closed string fluxes. Indeed this may be explicitly checked by plugging such backgrounds in the DBI+CS action for 7-branes, see [25]. In fact it is known that SUSY breaking imaginary self-dual (ISD) IIB 3-form fluxes correspond to giving a non-zero vacuum expectation value to the auxiliary fields of Kahler moduli [36]. So in order to see the effect of closed string fluxes we will work here with the effective action and plug non-vanishing vevs for these auxiliary fields.

In particular, in the context of Type IIB/F-theory compactifications a prominent role is played by the local Kahler modulus which is coupling to the stack of 7-branes. In a general Type IIB/F-theory compactification this Kahler modulus is the one among a number of such moduli which is relevant for the SUSY breaking soft terms, which will appear when . A good model for this structure is considering the CY manifold in ref.[37] with one small Kahler modulus and one big Kahler modulus with Kahler potential

(4.9)

with and . Here one takes and take both large so that the supergravity approximation is still valid. In the F-theory context the analogue of these moduli would correspond to the size of the 4-fold and the 6-fold respectively. For chiral matter fields living at F-theory matter curves one expects a behavior for the Kahler metrics in the dilute flux limit [38]

(4.10)

On the other hand if the Higgs doublets in matter curves are not chiral, they behave like scalars in a hypermultiplet, very much like in the previous case of open string fluxes. Under these conditions one expects kinetic terms for the Higgs multiplets of the form

(4.11)

This type of Higgs kinetic terms proportional to have been discussed in the past in the context of heterotic orbifold compactifications with subsectors in the untwisted spectrum and they display a shift symmetry under [40, 41]. Heterotic Type I S-duality indicates that such structure should also be present in Type IIB orientifolds. Recently Hebecker, Knochel and Weigand [42] have proposed that this shift symmetry may be at the origin of the tan boundary condition and studied its appearance also in Type II vacua. In our context the assumption of T-modulus dominance SUSY breaking allows to explicitly compute the relevant soft terms. Indeed applying standard supergavity formulae [43] one obtains for the Higgs mass parameters

(4.12)

so that

(4.13)

where is the gaugino mass, with the auxiliary field in the chiral multiplet. Now, unlike the open string flux case, the diagonal masses have both a SUSY contribution and a SUSY-breaking contribution and there is automatically a massless Higgs boson. We again obtain tan at the unification scale, this time automatically due to the mentioned shift symmetry. This value is however renormalized, as we point out below.

Figure 4: Structure of scales and the running of the gauge and Higgs coupling constants in this scheme.

As a general conclusion, we see that in string theory models in which the Higgs sector corresponds to a subsector with sitting in a hypermultiplet (before SUSY breaking), the condition is naturally obtained. In addition off-diagonal terms may be induced both by effects from open and closed string fluxes.

Let us finally comment on possible generalizations of this minimal Higgs structure. A first generalization is starting with sets of Higgs particles above . In that case minimal landscape fine-tuning will still prefer that only one combination of the Higgs scalars remains massless. Depending on how the original Higgs multiplets coupled to the different families the resulting Yukawas could inherit an interesting flavor structure. Another possible extension could be to dispose of R-parity in the initial MSSM spectrum since L/B-violating dimension four operators are suppressed due to the large mass of sfermions. In this case the Higgs could mix with sleptons . However in this case the approximate symmetry under the exchange of and would typically be absent and the prediction tan would be in danger, so this particular bilinear should be slightly supressed.

5 Higgs mass fine-tuning

We have seen how one may naturally obtain a massless Higgs with tan in string theory and, in particular, also in the context of Type IIB constructions with mass terms induced by open and closed string fluxes. In general one has to fine-tune the parameter with the Higgs masses . This may be done e.g. by partially canceling the contributions to from open and closed string fluxes.

However the above results are subject to loop corrections which will force to a redefinition of the fine-tuning. If the SUSY breaking scale is of order GeV the fine-tuning should be done to at least 4-loop order to cope with a hierarchy of nine orders of magnitude down to the EW scale. Still the idea is that even after these further fine-tuning corrections the Higgs scalar which remains light is approximately the one corresponding to the combination , i.e. that approximately tan.

In fact, if we assume that tan (as e.g. in eqs.(4.13)) before any loop correction is included, we know that the running of the parameters in between the scales and will renormalize tan. We expect that the large top quark Yukawa coupling will make after loop corrections. We also expect to obtain one massive Higgs eigenstate and a second one slightly tachyonic. This may be compensated to get a massless Higgs boson at this level by tuning with an open or closed string flux as explained in the previous section. Still, after this fine-tuning, the value of tan is no longer 1 but is given by tan, as explained in the previous section. In addition to this there will be higher order finite loop corrections which are expected to yield smaller negligible contributions to tan. So a good estimation of tan at the scale should be given by taking into account the running of the parameters in between and .

To compute the value of tan at we have to consider the RGE for the MSSM parameters in the region . In the present case we know that with a single Higgs field at the electroweak scale only the top-quark Yukawa coupling is relevant in this equation. Fortunately, the one-loop RGE in the limit were solved analytically in ref.[44] for the case of universal soft terms, i.e. as in the CMSSM model, which should be more than enough to evaluate this renormalization effect. One has tan with

(5.1)
(5.2)

where are the standard universal CMSSM parameters at the unification scale , and are known functions of the top Yukawa coupling and the three SM gauge coupling constants. For completeness these functions are provided in Appendix B. Note that in order to compute tan we do not need to know how runs since its value is fixed by the fine-tuning condition at .

Figure 5: Renormalization of tan in the region as a function of for three values of the top quark mass.

These functions involve integrals of coupling constants over the region to . There is an explicit dependence on the universal soft terms which are all of order but the results are quite insensitive to the precise values of those parameters. For definiteness we have computed tan for the boundary conditions , , , with , corresponding to the modulus dominance SUSY-breaking soft terms described around eq.(4.12), see e.g. [39]. In figure 5 we show the dependence of tan as a function of , where is taken as in section 3, from the gauge coupling unification condition. One observes that for in the range GeV tan is only slightly increased to a value around tan, depending on the value of the top-quark mass. In fact using the above formulae one can expand in a power series of the square of the top Yukawa coupling to find

(5.3)

It seems then that the tree level value of tan is only slightly deformed away from 1 after loop corrections. As we said, higher loop effects required to do a fully consistent fine-tuning are not expected to spoil this conclusion. An analogous conclusion was reached in [42] using different methods.888These results remain unchanged in the case of a R-parity violating MSSM since the new B/L-violating couplings will only appear in the Higgs mass running at two loops. One can trivially extend the calculation to the case in which color triplets remain below with very similar results.

A natural question is whether the Higgs mass terms discussed above scan in the string landscape. These masses depend on the local value of string flux densities in the region in the compact dimensions where the SM fields are localized. These local densities are in general not quantized, it is their integrals over 3- and 2-cycles which are quantized. As is well known in a generic compactification there may be of order a hundred different quantized closed string fluxes which may be turned on. All of them in general may contribute to the cosmological constant and could play a role in its anthropic solution [45]. The required energy spacing for the c.c. constant is so minute that at least some of these fluxes should e.g. be combined with anti-D3-branes on CY-throats in order to be able to fine-tune the c.c. following the KKLT approach [46]. On the other hand only a selected number of fluxes affect the SM branes in the cycle . Again although these fluxes are quantized it is the density at the location of the 7-branes which is relevant. However varying the flux quanta one can also control this local density. So, indeed, it seems plausible that the subset of the fluxes going through will scan in the string landscape. It would be interesting to materialize in some detail this expectation.

6 Proton decay

As we already advanced with a unification scale as low as GeV there is a danger of dimension 6 operators giving rise to proton decay rates much faster than experiment. In standard field theory GUTs, the proton decay dim=6 operators obtained after integrating out the massive doublet of gauge bosons are [28]

(6.1)
(6.2)

The first operator arises from the exchange of the heavy gauge bosons with masses between two 10-plets whereas the second from the exchange between a 10-plet and a 5-plet. Experimentally, the Super-Kamiokande limit on the chanel gives an absolute lower limit years [47]. This corresponds to a bound on

(6.3)

A value GeV is 5 times smaller and that could pose a problem. In F-theory GUTs the same proton decay operators as above will appear, the difference now being that the symmetry is broken due to a hypercharge flux. Due to this fact the coefficients of the operators may change substantially, as we now discuss.

Indeed, considering proton decay in the context of F-theory SU(5) unification provides a new interesting mechanism to suppress proton decay. A microscopic computation of the above dimension 6 proton decay operator would involve first computing couplings of the form e.g. and then integrating out the massive doublet . The computation of such trilinear couplings is rather similar to the computation of Yukawa couplings, in the sense that it also involves a triple overlap of internal wavefunctions, namely

(6.4)

where now are the internal wavefunctions of the broken SU(5) bosons . These form a doublet of massive gauge bosons with quantum numbers .

In standard 4d GUTs, the value of such couplings does not depend on the vev of the Higgs in the of SU(5), and so it is exactly the same before and after SU(5) breaking (to leading order). Hence, one may extract the trilinear couplings like directly from the SU(5) Lagrangian as the strength by which SU(5) gauge bosons couple to chiral matter, namely .

Now, the key point for proton decay suppression in F-theory is the fact that the ingredient that triggers SU(5) breaking is not a vev for a scalar in the adjoint of SU(5), but the presence of the hypercharge flux along the GUT 4-cycle . The mass of the gauge bosons is given by

(6.5)

where measures the density of hypercharge flux (see appendix C), which we take constant for simplicity. The flux quantization condition implies that is quantized in (i.e., its integral over 2-cycles of is an integer), so that and indeed . Finding the wavefunctions in (6.4) involves solving a Dirac or Laplace equation for them, in which any flux threading will enter. We then have that both the wavefunctions for chiral fields and massive gauge bosons , depend on the internal fluxes on , and in particular on the hypercharge flux . As a result, adding an hypercharge flux will necessarily change the value of the effective 4d couplings (6.4): while in the absence of such couplings must be in its presence they will have a new value.

To show that this new value will be suppressed with respect to we need some machinery from wavefunction computation in F-theory GUT models. Here we will try to be schematic, referring the reader to appendix C and to [48] (see also [32, 33, 49, 50, 51]) for more details on the subject. In F-theory SU(5) models there are basically two kinds of wavefunctions: the ones that are peaked at the matter curves of , namely , and , and the ones that are spread all over the 4-cycle , namely the SU(5) gauge bosons and in particular . As they come from different sectors of the theory, these two kinds of wavefunctions feel the effect of the hypercharge flux in a different way.

Indeed, let us consider the wavefunctions involved in the coupling in (6.4). Solving for them in a local patch of and assuming that the 4-cycle is sufficiently large (see appendix C and [48] for more details) we have that

(6.6)
(6.7)

where stand for local complex coordinates of the 4-cycle , and we have assumed that matter curve supporting the chiral fields is given by . The hypercharge dependence of the wavefunction is encoded in the hypercharge value and in , so that for a non-vanishing particles with different hypercharge have different wavefunctions. Here , , and stand for densities of fluxes threading the 4-cycle , and in particular is the density of the flux necessary to have three families of 10’s along . The parameter stands for the slope of the intersection between the SU(5) 4-cycle and the U(1) 7-brane intersecting in . Such intersection scale is typically of the order of the fundamental scale of F-theory ( in a perturbative IIB orientifold), which implies that are highly peaked along the matter curve . Finally, is a three-dimensional vector that depends on and the flux densities, and the ’s are normalization factors that insure that such fields are canonically normalized.

Both and the quantities that appear in the exponential factor of are family independent: the only dependence of the family index corresponding to the power of (the matter curve coordinate) that appears in the wavefunction. It has been found [32, 33, 49, 50] that with this prescription (that assigns the power to the first family, etc.) one can reproduce the mass hierarchy between families observed in nature.

Notice that the fact that , and are non-zero gives a gaussian profile to these wavefuctions, and this allows to carry the integral for by replacing with . This is important since otherwise we would need geometrical information about the full manifold , which is in general not available. Notice also that the wavefunction for the boson is only affected by the hypercharge flux density , and that in the limit we recover a constant wavefunction. This is to be expected, since at this limit the SU(5) symmetry is restored and become massless gauge bosons, which always have a constant profile.

Given these facts we are now ready to compute the coupling above. First notice that in the limit the integral is trivial in the sense that is constant, since

(6.8)

where used that for , the normalization factor is simply . Hence in this limit we recover the result expected from SU(5) gauge invariance.

This result is no longer true when and so the wavefunction has a non-trivial profile. Then one finds that there is a suppression in the above coupling which is family dependent, and bigger for lower families. Indeed, to get an estimate of this coupling it is useful to take the approximation and treat the Gaussian profile exp as a -function in the coordinate , which is nothing but asking that the matter wavefunctions are fully localized in . That is, we take the limit in which